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%TCIDATA{Created=Sunday, November 21, 1999 18:11:39}
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\input{tcilatex}
\begin{document}


\section{Exam}

\subsubsection{Comment}

I detta exempel testar vi studentens kunskaper i stokastiska variabler.

\subsubsection{Text}

\section{Stokastiska variabler}

Eftersom uppgifterna r\"{a}ttas av en dator \"{a}r det viktigt att du
skriver p\aa\ samma s\"{a}tt som i boken.

M\"{a}ngder skrivs $\left\{ {}\right\} $ CTRL+5. Elementen skrivs $%
a_{1},a_{2},a_{3},\ldots $ d\"{a}r $\ldots $ \"{a}r CTRL+ldots inte tre
punkter.

\subsection{Comment}

seed:=12345

\subsection{Setup}

Errors: report

Choices: No Break

Title: Probability

Submit:Click to Grade

$\limfunc{nplaces}(x,n)=1.0\left\lfloor 10^{n}x+0.5\right\rfloor /10^{n}$

\section{Part}

\section{Text}

\section{Problemdel}

\section{Question}

\subsection{Comment}

Allm\"{a}na diskreta f\"{o}rdelningar

\subsection{Variant}

\subsubsection{Setup}

$p_{1}:=\limfunc{nplaces}(\func{rand}(700,900)/1000,2)$

$p_{2}:=\limfunc{nplaces}(\func{rand}(500,800)/1000,2)$

$p_{3}:=\limfunc{nplaces}(\func{rand}(400,600)/1000,2)$

$s_{0}:=\limfunc{nplaces}\left( \left( 1-p_{1}\right) \left( 1-p_{2}\right)
\left( 1-p_{3}\right) ,2\right) $

$s_{1}:=\limfunc{nplaces}\left( p_{1}\left( 1-p_{2}\right) \left(
1-p_{3}\right) +\left( 1-p_{1}\right) p_{2}\left( 1-p_{3}\right) +\left(
1-p_{1}\right) \left( 1-p_{2}\right) p_{3},2\right) $

$s_{2}:=\limfunc{nplaces}\left( p_{1}p_{2}\left( 1-p_{3}\right) +p_{1}\left(
1-p_{2}\right) p_{3}+\left( 1-p_{1}\right) p_{2}p_{3},2\right) $

$s_{3}:=\limfunc{nplaces}\left( p_{1}p_{2}p_{3},2\right) $

$\limfunc{svar}:=\left( s_{0},s_{1},s_{2},s_{3}\right) $

\subsubsection{Statement}

Till en dator \"{a}r tre terminaler kopplade. Terminalerna anv\"{a}nds
oberoende av varandra och sannolikheten f\"{o}r att de i ett givet \"{o}%
gonblick skall anv\"{a}ndas \"{a}r $%
%TCIMACRO{\FORMULA{p_{1}}{p_{1}}{evaluate}}%
%BeginExpansion
p_{1}%
%EndExpansion
$, $%
%TCIMACRO{\FORMULA{p_{2}}{p_{2}}{evaluate}}%
%BeginExpansion
p_{2}%
%EndExpansion
$ och $%
%TCIMACRO{\FORMULA{p_{3}}{p_{3}}{evaluate}}%
%BeginExpansion
p_{3}%
%EndExpansion
$. Best\"{a}m sannolikhetsfunktionen f\"{o}r antalet terminaler som \"{a}r i
bruk i ett givet \"{o}gonblick (svara med 2 decimaler).

\subsubsection{Substatement}

$P\left( X=0\right) =$

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{1}$)

\subsubsection{Substatement}

$P\left( X=1\right) =$

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{2}$)

\subsubsection{Substatement}

$P\left( X=2\right) =$

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{3}$)

\subsubsection{Substatement}

$P\left( X=3\right) =$

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{4}$)

\paragraph{Solution}

S\"{a}tt 
\begin{align*}
X& =\text{antalet terminaler i bruk vid ett givet \"{o}gonblick} \\
T_{1}& =\text{terminal 1 \"{a}r i bruk} \\
T_{2}& =\text{terminal 2 \"{a}r i bruk} \\
T_{3}& =\text{terminal 3 \"{a}r i bruk }
\end{align*}%
Det g\"{a}ller%
\begin{align*}
P\left( X=0\right) & =P\left( \complement T_{1}\cap \complement T_{2}\cap
\complement T_{3}\right) \\
\left\{ \text{oberoende}\right\} & =%
%TCIMACRO{\FORMULA{1-p_{1}}{1-p_{1}}{evaluate}}%
%BeginExpansion
1-p_{1}%
%EndExpansion
%TCIMACRO{\FORMULA{1-p_{2}}{1-p_{2}}{evaluate}}%
%BeginExpansion
1-p_{2}%
%EndExpansion
%TCIMACRO{\FORMULA{1-p_{3}}{1-p_{3}}{evaluate}}%
%BeginExpansion
1-p_{3}%
%EndExpansion
\frac{1}{4}\frac{1}{3}\frac{1}{2} \\
& =%
%TCIMACRO{%
%\FORMULA{\left( 1-p_{1}\right) \left( 1-p_{2}\right) \left( 1-p_{3}\right) }{-\left( p_{1}-1\right) \left( p_{2}-1\right) \left( p_{3}-1\right) }{evaluate}}%
%BeginExpansion
-\left( p_{1}-1\right) \left( p_{2}-1\right) \left( p_{3}-1\right) %
%EndExpansion
\frac{1}{24} \\
P\left( X=1\right) & =P\left( T_{1}\cap \complement T_{2}\cap \complement
T_{3}\right) \\
& +P\left( \complement T_{1}\cap T_{2}\cap \complement T_{3}\right) \\
& +P\left( \complement T_{1}\cap \complement T_{2}\cap T_{3}\right) \\
\left\{ \text{oberoende}\right\} & =%
%TCIMACRO{\FORMULA{p_{1}}{p_{1}}{evaluate}}%
%BeginExpansion
p_{1}%
%EndExpansion
%TCIMACRO{\FORMULA{1-p_{2}}{1-p_{2}}{evaluate}}%
%BeginExpansion
1-p_{2}%
%EndExpansion
%TCIMACRO{\FORMULA{1-p_{3}}{1-p_{3}}{evaluate}}%
%BeginExpansion
1-p_{3}%
%EndExpansion
\frac{3}{4}\frac{1}{3}\frac{1}{2}+%
%TCIMACRO{\FORMULA{1-p_{1}}{1-p_{1}}{evaluate}}%
%BeginExpansion
1-p_{1}%
%EndExpansion
%TCIMACRO{\FORMULA{p_{2}}{p_{2}}{evaluate}}%
%BeginExpansion
p_{2}%
%EndExpansion
%TCIMACRO{\FORMULA{1-p_{3}}{1-p_{3}}{evaluate}}%
%BeginExpansion
1-p_{3}%
%EndExpansion
\frac{1}{4}\frac{2}{3}\frac{1}{2}+%
%TCIMACRO{\FORMULA{1-p_{1}}{1-p_{1}}{evaluate}}%
%BeginExpansion
1-p_{1}%
%EndExpansion
%TCIMACRO{\FORMULA{1-p_{2}}{1-p_{2}}{evaluate}}%
%BeginExpansion
1-p_{2}%
%EndExpansion
%TCIMACRO{\FORMULA{p_{3}}{p_{3}}{evaluate}}%
%BeginExpansion
p_{3}%
%EndExpansion
\frac{1}{4}\frac{1}{3}\frac{1}{2} \\
& =%
%TCIMACRO{%
%\FORMULA{p_{1}\left( 1-p_{2}\right) \left( 1-p_{3}\right) +\left( 1-p_{1}\right) p_{2}\left( 1-p_{3}\right) +\left( 1-p_{1}\right) \left( 1-p_{2}\right) p_{3}}{p_{1}\left( p_{2}-1\right) \left( p_{3}-1\right) +p_{2}\left( p_{1}-1\right) \left( p_{3}-1\right) +\allowbreak p_{3}\left( p_{1}-1\right) \left( p_{2}-1\right) }{evaluate}}%
%BeginExpansion
p_{1}\left( p_{2}-1\right) \left( p_{3}-1\right) +p_{2}\left( p_{1}-1\right) \left( p_{3}-1\right) +\allowbreak p_{3}\left( p_{1}-1\right) \left( p_{2}-1\right) %
%EndExpansion
\frac{1}{4} \\
P\left( X=2\right) & =P\left( T_{1}\cap T_{2}\cap \complement T_{3}\right) \\
& +P\left( T_{1}\cap \complement T_{2}\cap T_{3}\right) \\
& +P\left( \complement T_{1}\cap T_{2}\cap T_{3}\right) \\
\left\{ \text{oberoende}\right\} & =%
%TCIMACRO{\FORMULA{p_{1}}{p_{1}}{evaluate}}%
%BeginExpansion
p_{1}%
%EndExpansion
%TCIMACRO{\FORMULA{p_{2}}{p_{2}}{evaluate}}%
%BeginExpansion
p_{2}%
%EndExpansion
%TCIMACRO{\FORMULA{1-p_{3}}{1-p_{3}}{evaluate}}%
%BeginExpansion
1-p_{3}%
%EndExpansion
\frac{3}{4}\frac{2}{3}\frac{1}{2}+%
%TCIMACRO{\FORMULA{p_{1}}{p_{1}}{evaluate}}%
%BeginExpansion
p_{1}%
%EndExpansion
%TCIMACRO{\FORMULA{1-p_{2}}{1-p_{2}}{evaluate}}%
%BeginExpansion
1-p_{2}%
%EndExpansion
%TCIMACRO{\FORMULA{p_{3}}{p_{3}}{evaluate}}%
%BeginExpansion
p_{3}%
%EndExpansion
\frac{3}{4}\frac{1}{3}\frac{1}{2}+%
%TCIMACRO{\FORMULA{1-p_{1}}{1-p_{1}}{evaluate}}%
%BeginExpansion
1-p_{1}%
%EndExpansion
%TCIMACRO{\FORMULA{p_{2}}{p_{2}}{evaluate}}%
%BeginExpansion
p_{2}%
%EndExpansion
%TCIMACRO{\FORMULA{p_{3}}{p_{3}}{evaluate}}%
%BeginExpansion
p_{3}%
%EndExpansion
\frac{1}{4}\frac{2}{3}\frac{1}{2} \\
& =%
%TCIMACRO{%
%\FORMULA{p_{1}p_{2}\left( 1-p_{3}\right) +p_{1}\left( 1-p_{2}\right) p_{3}+\left( 1-p_{1}\right) p_{2}p_{3}}{-p_{1}p_{2}\left( p_{3}-1\right) -p_{1}p_{3}\left( p_{2}-1\right) -p_{2}p_{3}\left( p_{1}-1\right) \allowbreak }{evaluate}}%
%BeginExpansion
-p_{1}p_{2}\left( p_{3}-1\right) -p_{1}p_{3}\left( p_{2}-1\right) -p_{2}p_{3}\left( p_{1}-1\right) \allowbreak %
%EndExpansion
\frac{11}{24} \\
P\left( X=3\right) & =P\left( T_{1}\cap T_{2}\cap T_{3}\right) \\
\left\{ \text{oberoende}\right\} & =%
%TCIMACRO{\FORMULA{p_{1}}{p_{1}}{evaluate}}%
%BeginExpansion
p_{1}%
%EndExpansion
%TCIMACRO{\FORMULA{p_{2}}{p_{2}}{evaluate}}%
%BeginExpansion
p_{2}%
%EndExpansion
%TCIMACRO{\FORMULA{p_{3}}{p_{3}}{evaluate}}%
%BeginExpansion
p_{3}%
%EndExpansion
\frac{3}{4}\frac{2}{3}\frac{1}{2} \\
& =%
%TCIMACRO{\FORMULA{p_{1}p_{2}p_{3}}{p_{1}p_{2}p_{3}}{evaluate}}%
%BeginExpansion
p_{1}p_{2}p_{3}%
%EndExpansion
\frac{1}{4}
\end{align*}

\subsection{Variant}

\subsubsection{Setup}

$p_{1}:=\limfunc{nplaces}(\func{rand}(95,99)/100,2)$

$p_{2}:=\limfunc{nplaces}(\func{rand}(95,99)/100,2)$

$p_{3}:=\limfunc{nplaces}(\func{rand}(85,95)/100,2)$

$q_{0}:=\limfunc{nplaces}(\left( 1-p_{1}\right) \left( 1-p_{2}\right) \left(
1-p_{3}\right) ,3)$

$q_{10}:=\limfunc{nplaces}(q_{0}+p_{1}\left( 1-p_{2}\right) \left(
1-p_{3}\right) +\left( 1-p_{1}\right) p_{2}\left( 1-p_{3}\right) ,3)$

$q_{20}:=\limfunc{nplaces}(q_{10}+\left( 1-p_{1}\right) \left(
1-p_{2}\right) p_{3}+p_{1}p_{2}\left( 1-p_{3}\right) ,3)$

$q_{30}:=\limfunc{nplaces}(q_{20}+p_{1}\left( 1-p_{2}\right) p_{3}+\left(
1-p_{1}\right) p_{2}p_{3},3)$

$q_{40}:=\limfunc{nplaces}(q_{30}+p_{1}p_{2}p_{3},3)$

$a_{0}:=\limfunc{nplaces}(\left( 1-p_{1}\right) \left( 1-p_{2}\right) \left(
1-p_{3}\right) ,3)$

$a_{1}:=\limfunc{nplaces}(a_{0}+p_{1}\left( 1-p_{2}\right) \left(
1-p_{3}\right) +\left( 1-p_{1}\right) p_{2}\left( 1-p_{3}\right) +\left(
1-p_{1}\right) \left( 1-p_{2}\right) p_{3},3)$

$a_{2}:=\limfunc{nplaces}(a_{1}+p_{1}p_{2}\left( 1-p_{3}\right) +p_{1}\left(
1-p_{2}\right) p_{3}+\left( 1-p_{1}\right) p_{2}p_{3},3)$

$a_{3}:=\limfunc{nplaces}(a_{2}+p_{1}p_{2}p_{3},3)$

$\limfunc{svar}:=\left( \left\{ 0,10,20,30,40\right\} ,\left\{
0,1,2,3\right\} ,\left\{ q_{0},q_{10},q_{20},q_{30},q_{40}\right\} ,\left\{
a_{0},a_{1},a_{2},a_{3}\right\} \right) $

\subsubsection{Statement}

Man har tv\aa\ vattenkraftverk med effekten $10$ MW var och ett v\"{a}%
rmekraftverk med effekten $20$ MW. Kraftverken g\aa r s\"{o}nder oberoende
av varandra och sannolikheten f\"{o}r att de fungerar vid en viss tidpunkt 
\"{a}r $%
%TCIMACRO{\FORMULA{p_{1}}{p_{1}}{evaluate}}%
%BeginExpansion
p_{1}%
%EndExpansion
$, $%
%TCIMACRO{\FORMULA{p_{2}}{p_{2}}{evaluate}}%
%BeginExpansion
p_{2}%
%EndExpansion
$ och $%
%TCIMACRO{\FORMULA{p_{3}}{p_{3}}{evaluate}}%
%BeginExpansion
p_{3}%
%EndExpansion
$. Bilda de stokastiska variablerna%
\begin{align*}
X& =\text{tillg\"{a}nglig effekt i MW,} \\
Y& =\text{antal kraftstationer i bruk.}
\end{align*}

\subsubsection{Substatement}

Ange utfallsrummet f\"{o}r $X$ p\aa\ formen $\left\{ a,b,\ldots ,c\right\} $.

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{1}$)

\subsubsection{Substatement}

Ange utfallsrummet f\"{o}r $Y$ p\aa\ formen $\left\{ a,b,\ldots ,c\right\} $.

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{2}$)

\subsubsection{Substatement}

Ange f\"{o}rdelningsfunktionen f\"{o}r $X$ p\aa\ formen $\left\{
p_{1},\ldots ,p_{k}\right\} $ (tre decimaler).

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{3}$)

\subsubsection{Substatement}

Ange f\"{o}rdelningsfunktionen f\"{o}r $Y$ p\aa\ formen $\left\{
p_{1},\ldots ,p_{k}\right\} $ (tre decimaler).

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{4}$)

\paragraph{Solution}

F\"{o}r utfallsrummen finner vi%
\begin{align*}
\Omega _{X}& =\left\{ 0,10,20,30,40\right\} \\
\Omega _{Y}& =\left\{ 0,1,2,3\right\}
\end{align*}%
motsvarande sannolikhetsfunktioner blir%
\begin{equation*}
P\left( X=x\right) =\left\{ 
\begin{array}{ccc}
%TCIMACRO{\FORMULA{\left( 1-p_{1}\right) }{1-p_{1}}{evaluate}}%
%BeginExpansion
1-p_{1}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{\left( 1-p_{2}\right) }{1-p_{2}}{evaluate}}%
%BeginExpansion
1-p_{2}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{\left( 1-p_{3}\right) }{1-p_{3}}{evaluate} }%
%BeginExpansion
1-p_{3}
%EndExpansion
& \text{om} & x=0 \\ 
%TCIMACRO{\FORMULA{p_{1}}{p_{1}}{evaluate}}%
%BeginExpansion
p_{1}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{\left( 1-p_{2}\right) }{1-p_{2}}{evaluate}}%
%BeginExpansion
1-p_{2}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{\left( 1-p_{3}\right) }{1-p_{3}}{evaluate}}%
%BeginExpansion
1-p_{3}%
%EndExpansion
+%
%TCIMACRO{\FORMULA{\left( 1-p_{1}\right) }{1-p_{1}}{evaluate}}%
%BeginExpansion
1-p_{1}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{p_{2}}{p_{2}}{evaluate}}%
%BeginExpansion
p_{2}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{\left( 1-p_{3}\right) }{1-p_{3}}{evaluate} }%
%BeginExpansion
1-p_{3}
%EndExpansion
& \text{om} & x=10 \\ 
%TCIMACRO{\FORMULA{\left( 1-p_{1}\right) }{1-p_{1}}{evaluate}}%
%BeginExpansion
1-p_{1}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{\left( 1-p_{2}\right) }{1-p_{2}}{evaluate}}%
%BeginExpansion
1-p_{2}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{p_{3}}{p_{3}}{evaluate}}%
%BeginExpansion
p_{3}%
%EndExpansion
+%
%TCIMACRO{\FORMULA{p_{1}}{p_{1}}{evaluate}}%
%BeginExpansion
p_{1}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{p_{2}}{p_{2}}{evaluate}}%
%BeginExpansion
p_{2}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{\left( 1-p_{3}\right) }{1-p_{3}}{evaluate} }%
%BeginExpansion
1-p_{3}
%EndExpansion
& \text{om} & x=20 \\ 
%TCIMACRO{\FORMULA{p_{1}}{p_{1}}{evaluate}}%
%BeginExpansion
p_{1}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{\left( 1-p_{2}\right) }{1-p_{2}}{evaluate}}%
%BeginExpansion
1-p_{2}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{p_{3}}{p_{3}}{evaluate}}%
%BeginExpansion
p_{3}%
%EndExpansion
+%
%TCIMACRO{\FORMULA{\left( 1-p_{1}\right) }{1-p_{1}}{evaluate}}%
%BeginExpansion
1-p_{1}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{p_{2}}{p_{2}}{evaluate}}%
%BeginExpansion
p_{2}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{p_{3}}{p_{3}}{evaluate} }%
%BeginExpansion
p_{3}
%EndExpansion
& \text{om} & x=30 \\ 
%TCIMACRO{\FORMULA{p_{1}}{p_{1}}{evaluate}}%
%BeginExpansion
p_{1}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{p_{2}}{p_{2}}{evaluate}}%
%BeginExpansion
p_{2}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{p_{3}}{p_{3}}{evaluate}}%
%BeginExpansion
p_{3}%
%EndExpansion
p_{1}p_{2}p_{3} & \text{om} & x=40%
\end{array}%
\right. =\left\{ 
\begin{array}{ccc}
%TCIMACRO{\FORMULA{q_{0}}{q_{0}}{evaluate} }%
%BeginExpansion
q_{0}
%EndExpansion
& \text{om} & x=0 \\ 
%TCIMACRO{\FORMULA{q_{10}-q_{0}}{q_{10}-q_{0}}{evaluate} }%
%BeginExpansion
q_{10}-q_{0}
%EndExpansion
& \text{om} & x=10 \\ 
%TCIMACRO{\FORMULA{q_{20}-q_{10}}{q_{20}-q_{10}}{evaluate} }%
%BeginExpansion
q_{20}-q_{10}
%EndExpansion
& \text{om} & x=20 \\ 
%TCIMACRO{\FORMULA{q_{30}-q_{20}}{q_{30}-q_{20}}{evaluate} }%
%BeginExpansion
q_{30}-q_{20}
%EndExpansion
& \text{om} & x=30 \\ 
%TCIMACRO{\FORMULA{q_{40}-q_{30}}{q_{40}-q_{30}}{evaluate} }%
%BeginExpansion
q_{40}-q_{30}
%EndExpansion
& \text{om} & x=40%
\end{array}%
\right.
\end{equation*}%
samt%
\begin{equation*}
P\left( Y=y\right) =\left\{ 
\begin{array}{ccc}
%TCIMACRO{\FORMULA{\left( 1-p_{1}\right) }{1-p_{1}}{evaluate}}%
%BeginExpansion
1-p_{1}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{\left( 1-p_{2}\right) }{1-p_{2}}{evaluate}}%
%BeginExpansion
1-p_{2}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{\left( 1-p_{3}\right) }{1-p_{3}}{evaluate} }%
%BeginExpansion
1-p_{3}
%EndExpansion
& \text{om} & y=0 \\ 
%TCIMACRO{\FORMULA{p_{1}}{p_{1}}{evaluate}}%
%BeginExpansion
p_{1}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{\left( 1-p_{2}\right) }{1-p_{2}}{evaluate}}%
%BeginExpansion
1-p_{2}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{\left( 1-p_{3}\right) }{1-p_{3}}{evaluate}}%
%BeginExpansion
1-p_{3}%
%EndExpansion
+%
%TCIMACRO{\FORMULA{\left( 1-p_{1}\right) }{1-p_{1}}{evaluate}}%
%BeginExpansion
1-p_{1}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{p_{2}}{p_{2}}{evaluate}}%
%BeginExpansion
p_{2}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{\left( 1-p_{3}\right) }{1-p_{3}}{evaluate}}%
%BeginExpansion
1-p_{3}%
%EndExpansion
+%
%TCIMACRO{\FORMULA{\left( 1-p_{1}\right) }{1-p_{1}}{evaluate}}%
%BeginExpansion
1-p_{1}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{\left( 1-p_{2}\right) }{1-p_{2}}{evaluate}}%
%BeginExpansion
1-p_{2}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{p_{3}}{p_{3}}{evaluate} }%
%BeginExpansion
p_{3}
%EndExpansion
& \text{om} & y=1 \\ 
%TCIMACRO{\FORMULA{p_{1}}{p_{1}}{evaluate}}%
%BeginExpansion
p_{1}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{p_{2}}{p_{2}}{evaluate}}%
%BeginExpansion
p_{2}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{\left( 1-p_{3}\right) }{1-p_{3}}{evaluate}}%
%BeginExpansion
1-p_{3}%
%EndExpansion
+%
%TCIMACRO{\FORMULA{p_{1}}{p_{1}}{evaluate}}%
%BeginExpansion
p_{1}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{\left( 1-p_{2}\right) }{1-p_{2}}{evaluate}}%
%BeginExpansion
1-p_{2}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{p_{3}}{p_{3}}{evaluate}}%
%BeginExpansion
p_{3}%
%EndExpansion
+%
%TCIMACRO{\FORMULA{\left( 1-p_{1}\right) }{1-p_{1}}{evaluate}}%
%BeginExpansion
1-p_{1}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{p_{2}}{p_{2}}{evaluate}}%
%BeginExpansion
p_{2}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{p_{3}}{p_{3}}{evaluate} }%
%BeginExpansion
p_{3}
%EndExpansion
& \text{om} & y=2 \\ 
%TCIMACRO{\FORMULA{p_{1}}{p_{1}}{evaluate}}%
%BeginExpansion
p_{1}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{p_{2}}{p_{2}}{evaluate}}%
%BeginExpansion
p_{2}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{p_{3}}{p_{3}}{evaluate} }%
%BeginExpansion
p_{3}
%EndExpansion
& \text{om} & y=3%
\end{array}%
\right. =\left\{ 
\begin{array}{ccc}
%TCIMACRO{\FORMULA{a_{0}}{a_{0}}{evaluate} }%
%BeginExpansion
a_{0}
%EndExpansion
& \text{om} & y=0 \\ 
%TCIMACRO{\FORMULA{a_{1}-a_{0}}{a_{1}-a_{0}}{evaluate} }%
%BeginExpansion
a_{1}-a_{0}
%EndExpansion
& \text{om} & y=1 \\ 
%TCIMACRO{\FORMULA{a_{2}-a_{1}}{a_{2}-a_{1}}{evaluate} }%
%BeginExpansion
a_{2}-a_{1}
%EndExpansion
& \text{om} & y=2 \\ 
%TCIMACRO{\FORMULA{a_{3}-a_{2}}{a_{3}-a_{2}}{evaluate} }%
%BeginExpansion
a_{3}-a_{2}
%EndExpansion
& \text{om} & y=3%
\end{array}%
\right.
\end{equation*}

H\"{a}rav f\"{o}ljer att%
\begin{equation*}
P\left( X\leq x\right) =\left\{ 
\begin{array}{ccc}
%TCIMACRO{\FORMULA{q_{0}}{q_{0}}{evaluate} }%
%BeginExpansion
q_{0}
%EndExpansion
& \text{om} & x=0 \\ 
%TCIMACRO{\FORMULA{q_{10}}{q_{10}}{evaluate} }%
%BeginExpansion
q_{10}
%EndExpansion
& \text{om} & x\leq 10 \\ 
%TCIMACRO{\FORMULA{q_{20}}{q_{20}}{evaluate} }%
%BeginExpansion
q_{20}
%EndExpansion
& \text{om} & x\leq 20 \\ 
%TCIMACRO{\FORMULA{q_{30}}{q_{30}}{evaluate} }%
%BeginExpansion
q_{30}
%EndExpansion
& \text{om} & x\leq 30 \\ 
%TCIMACRO{\FORMULA{q_{40}}{q_{40}}{evaluate} }%
%BeginExpansion
q_{40}
%EndExpansion
& \text{om} & x\leq 40%
\end{array}%
\right.
\end{equation*}%
samt%
\begin{equation*}
P\left( Y=y\right) =\left\{ 
\begin{array}{ccc}
%TCIMACRO{\FORMULA{a_{0}}{a_{0}}{evaluate} }%
%BeginExpansion
a_{0}
%EndExpansion
& \text{om} & y=0 \\ 
%TCIMACRO{\FORMULA{a_{1}}{a_{1}}{evaluate} }%
%BeginExpansion
a_{1}
%EndExpansion
& \text{om} & y=1 \\ 
%TCIMACRO{\FORMULA{a_{2}}{a_{2}}{evaluate} }%
%BeginExpansion
a_{2}
%EndExpansion
& \text{om} & y=2 \\ 
%TCIMACRO{\FORMULA{a_{3}}{a_{3}}{evaluate} }%
%BeginExpansion
a_{3}
%EndExpansion
& \text{om} & y=3%
\end{array}%
\right. \text{.}
\end{equation*}

\subsection{Variant}

\subsubsection{Setup}

$p_{1}:=\limfunc{nplaces}(\func{rand}(15,25)/100,2)$

$p_{2}:=\limfunc{nplaces}(\func{rand}(25,35)/100,2)$

$p_{3}:=\limfunc{nplaces}(\func{rand}(10,100p_{1}+100p_{2})/100,2)$

$p_{4}:=p_{1}+p_{2}-p_{3}$

$\limfunc{svar}:=\left( 1-p_{3},p_{2}-p_{4},p_{1}-p_{4},p_{4}\right) $

\subsubsection{Statement}

Hos en viss typ av flygplan som skall tas in f\"{o}r \"{o}versyn kan tv\aa\ %
speciellt allvarliga fel, $A$ respektive $B$, f\"{o}rekomma. Om ett flygplan
har b\aa de fel $A$ och $B$ s\aa\ tar \"{o}versynen $5$ dagar, om det endast
har fel $A$ tar den $3$ dagar, om endast fel $B$ s\aa\ $2$ dagar, och om det
varken har fel $A$ eller $B$ s\aa\ tar den $1$ dag.

Sannolikheten att ett flygplan har \aa tminstone fel $A$ \"{a}r $%
%TCIMACRO{\FORMULA{p_{1}}{p_{1}}{evaluate}}%
%BeginExpansion
p_{1}%
%EndExpansion
$ och att det har \aa tminstone fel $B$ \"{a}r $%
%TCIMACRO{\FORMULA{p_{2}}{p_{2}}{evaluate}}%
%BeginExpansion
p_{2}%
%EndExpansion
$. Sannolikheten att det har varken fel $A$ eller $B$ \"{a}r $%
%TCIMACRO{\FORMULA{1-p_{3}}{1-p_{3}}{evaluate}}%
%BeginExpansion
1-p_{3}%
%EndExpansion
$. Best\"{a}m sannolikhetsf\"{o}rdelningen f\"{o}r den tid \"{o}versynen av
ett flygplan tar (svara med tre decimaler).

\subsubsection{Substatement}

$P\left( X=1\right) =$

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{1}$)

\subsubsection{Substatement}

$P\left( X=2\right) =$

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{2}$)

\subsubsection{Substatement}

$P\left( X=3\right) =$

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{3}$)

\subsubsection{Substatement}

$P\left( X=5\right) =$

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{4}$)

\paragraph{Solution}

S\"{a}tt 
\begin{align*}
X_{i}& =\text{\"{o}versynstiden f\"{o}r flygplan }i\quad i=1,2,\dots ,25 \\
X_{i}& \in \left\{ 1,2,3,5\right\}
\end{align*}%
d\aa\ erh\aa lls f\"{o}rdelningen 
\begin{align*}
P\left( X_{i}=1\right) & =P\left( \complement A\cap \complement B\right) =%
%TCIMACRO{\FORMULA{1-p_{3}}{1-p_{3}}{evaluate} }%
%BeginExpansion
1-p_{3}
%EndExpansion
\\
P\left( X_{i}=2\right) & =P\left( \complement A\cap B\right) =P\left(
B\right) -P\left( A\cap B\right) \\
& =%
%TCIMACRO{\FORMULA{p_{2}}{p_{2}}{evaluate}}%
%BeginExpansion
p_{2}%
%EndExpansion
-%
%TCIMACRO{\FORMULA{p_{4}}{p_{4}}{evaluate}}%
%BeginExpansion
p_{4}%
%EndExpansion
=%
%TCIMACRO{\FORMULA{p_{2}-p_{4}}{p_{2}-p_{4}}{evaluate} }%
%BeginExpansion
p_{2}-p_{4}
%EndExpansion
\\
P\left( X_{i}=3\right) & =P\left( A\cap \complement B\right) =P\left(
A\right) -P\left( A\cap B\right) \\
& =%
%TCIMACRO{\FORMULA{p_{1}}{p_{1}}{evaluate}}%
%BeginExpansion
p_{1}%
%EndExpansion
-%
%TCIMACRO{\FORMULA{p_{4}}{p_{4}}{evaluate}}%
%BeginExpansion
p_{4}%
%EndExpansion
=%
%TCIMACRO{\FORMULA{p_{1}-p_{4}}{p_{1}-p_{4}}{evaluate} }%
%BeginExpansion
p_{1}-p_{4}
%EndExpansion
\\
P\left( X_{i}=5\right) & =P\left( A\cap B\right) =%
%TCIMACRO{\FORMULA{p_{4}}{p_{4}}{evaluate}}%
%BeginExpansion
p_{4}%
%EndExpansion
\end{align*}

\subsection{Variant}

\subsubsection{Setup}

$p_{04}:=\limfunc{nplaces}(\func{rand}(5,15)/100,2)$

$p_{14}:=\limfunc{nplaces}(\func{rand}(25,35)/100,2)$

$p_{23}:=\limfunc{nplaces}(\func{rand}(35,45)/100,2)$

$p_{33}:=1-p_{04}-p_{14}-p_{23}$

$\limfunc{svar}:=\left( p_{04},p_{14}+p_{23},p_{33}\right) $

\subsubsection{Statement}

Den tv\aa dimensionella stokastiska variabeln $\left( X,Y\right) $ har den
simultana sannolikhetsfunktionen%
\begin{equation*}
P\left( X=i,Y=j\right) =\left\{ 
\begin{array}{ccc}
%TCIMACRO{\FORMULA{p_{33}}{p_{33}}{evaluate} }%
%BeginExpansion
p_{33}
%EndExpansion
& \text{om} & i=3,j=3 \\ 
%TCIMACRO{\FORMULA{p_{23}}{p_{23}}{evaluate} }%
%BeginExpansion
p_{23}
%EndExpansion
& \text{om} & i=2,j=3 \\ 
%TCIMACRO{\FORMULA{p_{14}}{p_{14}}{evaluate} }%
%BeginExpansion
p_{14}
%EndExpansion
& \text{om} & i=1,j=4 \\ 
%TCIMACRO{\FORMULA{p_{04}}{p_{04}}{evaluate} }%
%BeginExpansion
p_{04}
%EndExpansion
& \text{om} & i=0,j=4%
\end{array}%
\right.
\end{equation*}%
best\"{a}m sannolikhetsf\"{o}rdelningen f\"{o}r den stokastiska variabeln $%
X+Y$: (svara med tv\aa\ decimaler).

\subsubsection{Substatement}

$P\left( X=4\right) =$

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{1}$)

\subsubsection{Substatement}

$P\left( X=5\right) =$

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{2}$)

\subsubsection{Substatement}

$P\left( X=6\right) =$

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{3}$)

\paragraph{Solution}

Den stokastiska variabeln $Z=X+Y$ har utfallsrummet $\Omega _{Z}=\left\{
4,5,6\right\} $ och det g\"{a}ller%
\begin{equation*}
P\left( Z=k\right) =\left\{ 
\begin{array}{cc}
%TCIMACRO{\FORMULA{p_{04}}{p_{04}}{evaluate} }%
%BeginExpansion
p_{04}
%EndExpansion
& k=4 \\ 
%TCIMACRO{\FORMULA{p_{14}+p_{23}}{p_{14}+p_{23}}{evaluate} }%
%BeginExpansion
p_{14}+p_{23}
%EndExpansion
& k=5 \\ 
%TCIMACRO{\FORMULA{p_{33}}{p_{33}}{evaluate} }%
%BeginExpansion
p_{33}
%EndExpansion
& k=6%
\end{array}%
\right.
\end{equation*}

\section{Question}

\subsection{Comment}

Binomial- och hypergeometrisk f\"{o}rdelning

\subsection{Variant}

\subsubsection{Setup}

$p:=\limfunc{nplaces}(\func{rand}(10,90)/100,2)$

$n:=\func{rand}(5,20)$

$m_{1}:=\func{rand}(5,n)$

$m_{2}:=\func{rand}(2,n-2)$

$m_{3}:=\func{rand}(3,n-1)$

$m_{4}:=\func{rand}(m_{2}+2,n)$

$s_{1}:=\limfunc{nplaces}\left( \binom{n}{m_{1}}p^{m_{1}}\left( 1-p\right)
^{n-m_{1}},3\right) $

$s_{2}:=\limfunc{nplaces}\left( \sum_{k=0}^{m_{2}}\binom{n}{k}p^{k}\left(
1-p\right) ^{n-k},3\right) $

$s_{3}:=\limfunc{nplaces}\left( 1-\sum_{k=0}^{m_{3}-1}\binom{n}{k}%
p^{k}\left( 1-p\right) ^{n-k},3\right) $

$s_{4}:=\limfunc{nplaces}\left( \sum_{k=0}^{m_{4}-1}\binom{n}{k}p^{k}\left(
1-p\right) ^{n-k}-\sum_{k=0}^{m_{2}}\binom{n}{k}p^{k}\left( 1-p\right)
^{n-k},3\right) $

$\limfunc{svar}:=\left( s_{1},s_{2},s_{3},s_{4}\right) $

\subsubsection{Statement}

Givet en binomialf\"{o}rdelad stokastisk variabel $X$ med $n=%
%TCIMACRO{\FORMULA{n}{n}{evaluate}}%
%BeginExpansion
n%
%EndExpansion
$ och $p=%
%TCIMACRO{\FORMULA{p}{p}{evaluate}}%
%BeginExpansion
p%
%EndExpansion
$ best\"{a}m sannolikheterna (svara med tre decimalet.

\subsubsection{Substatement}

$P\left( X=%
%TCIMACRO{\FORMULA{m_{1}}{m_{1}}{evaluate}}%
%BeginExpansion
m_{1}%
%EndExpansion
\right) =$

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{1}$)

\subsubsection{Substatement}

$P\left( X\leq 
%TCIMACRO{\FORMULA{m_{2}}{m_{2}}{evaluate}}%
%BeginExpansion
m_{2}%
%EndExpansion
\right) =$

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{2}$)

\subsubsection{Substatement}

$P\left( X\geq 
%TCIMACRO{\FORMULA{m_{3}}{m_{3}}{evaluate}}%
%BeginExpansion
m_{3}%
%EndExpansion
\right) =$

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{3}$)

\subsubsection{Substatement}

$P\left( 
%TCIMACRO{\FORMULA{m_{2}+1}{m_{2}+1}{evaluate}}%
%BeginExpansion
m_{2}+1%
%EndExpansion
\leq X<%
%TCIMACRO{\FORMULA{m_{4}}{m_{4}}{evaluate}}%
%BeginExpansion
m_{4}%
%EndExpansion
\right) =$

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{4}$)

\paragraph{Solution}

F\"{o}ljande sannolikheter erh\aa lls

\begin{enumerate}
\item $P\left( X=%
%TCIMACRO{\FORMULA{m_{1}}{m_{1}}{evaluate}}%
%BeginExpansion
m_{1}%
%EndExpansion
\right) =%
%TCIMACRO{\FORMULA{s_{1}}{s_{1}}{evaluate}}%
%BeginExpansion
s_{1}%
%EndExpansion
$

\item $P\left( X\leq 
%TCIMACRO{\FORMULA{m_{2}}{m_{2}}{evaluate}}%
%BeginExpansion
m_{2}%
%EndExpansion
\right) =%
%TCIMACRO{\FORMULA{s_{2}}{s_{2}}{evaluate}}%
%BeginExpansion
s_{2}%
%EndExpansion
$

\item $P\left( X\geq 
%TCIMACRO{\FORMULA{m_{3}}{m_{3}}{evaluate}}%
%BeginExpansion
m_{3}%
%EndExpansion
\right) =%
%TCIMACRO{\FORMULA{s_{3}}{s_{3}}{evaluate}}%
%BeginExpansion
s_{3}%
%EndExpansion
$

\item $P\left( 
%TCIMACRO{\FORMULA{m_{3}}{m_{3}}{evaluate}}%
%BeginExpansion
m_{3}%
%EndExpansion
\leq X<%
%TCIMACRO{\FORMULA{m_{4}}{m_{4}}{evaluate}}%
%BeginExpansion
m_{4}%
%EndExpansion
\right) =%
%TCIMACRO{\FORMULA{s_{4}}{s_{4}}{evaluate}}%
%BeginExpansion
s_{4}%
%EndExpansion
$
\end{enumerate}

\subsection{Variant}

\subsubsection{Setup}

$p:=\limfunc{nplaces}(\func{rand}(20,60)/100,2)$

$n:=\func{rand}(9,50)$

$m:=\func{rand}(2,\frac{n}{3})$

$s:=\limfunc{nplaces}\left( 1-\sum_{k=0}^{m}\binom{n}{k}p^{k}\left(
1-p\right) ^{n-k},3\right) $

$\limfunc{svar}:=s$

\subsubsection{Statement}

I en datasal finns $%
%TCIMACRO{\FORMULA{n}{n}{evaluate}}%
%BeginExpansion
n%
%EndExpansion
$ datorer uppkopplade mot en server. Under $%
%TCIMACRO{%
%\FORMULA{\left\lfloor 100p\right\rfloor }{\left\lfloor 100p\right\rfloor }{evaluate}}%
%BeginExpansion
\left\lfloor 100p\right\rfloor %
%EndExpansion
$ $\%$ av tiden g\"{o}r dessa s\"{o}kningar i en databas. S\"{o}kningarna
intr\"{a}ffar oberoende av varandra. Vad \"{a}r sannolikheten att minst $%
%TCIMACRO{\FORMULA{m+1}{m+1}{evaluate}}%
%BeginExpansion
m+1%
%EndExpansion
$ s\"{o}kningar intr\"{a}ffar samtidigt vid en given tidpunkt?

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}$)

\paragraph{Solution}

S\"{a}tt 
\begin{equation*}
X=\text{antal s\"{o}kningar vid en given tidpunkt}
\end{equation*}%
d\aa\ g\"{a}ller att 
\begin{equation*}
X\in Bin\left( 
%TCIMACRO{\FORMULA{n}{n}{evaluate}}%
%BeginExpansion
n%
%EndExpansion
,%
%TCIMACRO{\FORMULA{p}{p}{evaluate}}%
%BeginExpansion
p%
%EndExpansion
\right)
\end{equation*}%
Den s\"{o}kta sannolikheten kan nu skrivas 
\begin{equation*}
P\left( X\geq 
%TCIMACRO{\FORMULA{m+1}{m+1}{evaluate}}%
%BeginExpansion
m+1%
%EndExpansion
\right) =1-\sum_{k=0}^{%
%TCIMACRO{\FORMULA{m}{m}{evaluate}}%
%BeginExpansion
m%
%EndExpansion
}\binom{%
%TCIMACRO{\FORMULA{n}{n}{evaluate}}%
%BeginExpansion
n%
%EndExpansion
}{k}%
%TCIMACRO{\FORMULA{p}{p}{evaluate}}%
%BeginExpansion
p%
%EndExpansion
^{k}\left( 1-%
%TCIMACRO{\FORMULA{p}{p}{evaluate}}%
%BeginExpansion
p%
%EndExpansion
\right) ^{%
%TCIMACRO{\FORMULA{n}{n}{evaluate}}%
%BeginExpansion
n%
%EndExpansion
-k}=%
%TCIMACRO{\FORMULA{s}{s}{evaluate}}%
%BeginExpansion
s%
%EndExpansion
\end{equation*}

\subsection{Variant}

\subsubsection{Setup}

$p:=\limfunc{nplaces}(\func{rand}(1,5)/100,2)$

$n:=\func{rand}(20,40)$

$\limfunc{svar}:=\limfunc{nplaces}\left( \left( 1-p\right) ^{n},3\right) $

\subsubsection{Statement}

En elektronisk produkt best\aa r av $%
%TCIMACRO{\FORMULA{n}{n}{evaluate}}%
%BeginExpansion
n%
%EndExpansion
$ stycken seriekopplade komponenter d\"{a}r varje komponent har
sannolikheten $%
%TCIMACRO{\FORMULA{p}{p}{evaluate}}%
%BeginExpansion
p%
%EndExpansion
$ att g\aa\ s\"{o}nder under en dag. Vad \"{a}r sannolikheten att produkten
fungerar vid dagens slut (svara med tre decimaler)?

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}$)

\paragraph{Solution}

S\"{a}tt%
\begin{equation*}
X=\text{antal defekta komponenter}
\end{equation*}%
d\aa\ g\"{a}ller att $X\in Bin\left( 
%TCIMACRO{\FORMULA{n}{n}{evaluate}}%
%BeginExpansion
n%
%EndExpansion
,%
%TCIMACRO{\FORMULA{p}{p}{evaluate}}%
%BeginExpansion
p%
%EndExpansion
\right) $. S\"{o}kt sannolikhet blir%
\begin{align*}
P\left( X=0\right) & =\binom{%
%TCIMACRO{\FORMULA{n}{n}{evaluate}}%
%BeginExpansion
n%
%EndExpansion
}{0}%
%TCIMACRO{\FORMULA{p}{p}{evaluate}}%
%BeginExpansion
p%
%EndExpansion
^{0}%
%TCIMACRO{\FORMULA{1-p}{1-p}{evaluate}}%
%BeginExpansion
1-p%
%EndExpansion
^{%
%TCIMACRO{\FORMULA{n}{n}{evaluate}}%
%BeginExpansion
n%
%EndExpansion
-0} \\
& =%
%TCIMACRO{\FORMULA{\limfunc{svar}}{\limfunc{svar}}{evaluate}}%
%BeginExpansion
\limfunc{svar}%
%EndExpansion
\end{align*}

\subsection{Variant}

\subsubsection{Setup}

$p:=\limfunc{nplaces}(\func{rand}(5,15)/1000,3)$

$n:=\func{rand}(90,110)$

$m:=\func{rand}(2,4)$

$l:=\func{rand}(600,1000)$

$\limfunc{svar}:=\limfunc{nplaces}\left( \sum_{k=0}^{m}\binom{n}{k}%
p^{k}\left( 1-p\right) ^{n-k},2\right) $

\subsubsection{Statement}

En kabel best\aa r av $%
%TCIMACRO{\FORMULA{n}{n}{evaluate}}%
%BeginExpansion
n%
%EndExpansion
$ st\aa ltr\aa dar och f\"{o}r var och en av dessa tr\aa dar g\"{a}ller att
sannolikheten f\"{o}r att en tr\aa d av $%
%TCIMACRO{\FORMULA{l}{l}{evaluate}}%
%BeginExpansion
l%
%EndExpansion
$ m:s l\"{a}ngd skall vara defekt \"{a}r $%
%TCIMACRO{\FORMULA{p}{p}{evaluate}}%
%BeginExpansion
p%
%EndExpansion
$. F\"{o}r att en kabel p\aa\ $%
%TCIMACRO{\FORMULA{l}{l}{evaluate}}%
%BeginExpansion
l%
%EndExpansion
$ m skall kunna b\"{a}ra angiven tyngd f\aa r h\"{o}gst $%
%TCIMACRO{\FORMULA{m}{m}{evaluate}}%
%BeginExpansion
m%
%EndExpansion
$ av tr\aa darna vara defekta. Vad \"{a}r sannolikheten f\"{o}r att $%
%TCIMACRO{\FORMULA{l}{l}{evaluate}}%
%BeginExpansion
l%
%EndExpansion
$ m kabel skall h\aa lla angiven tyngd (svara med tv\aa\ decimaler)?

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}$)

\paragraph{Solution}

S\"{a}tt%
\begin{equation*}
X=\text{antal defekta st\aa ltr\aa dar}
\end{equation*}%
d\aa\ g\"{a}ller att $X\in Bin\left( 
%TCIMACRO{\FORMULA{n}{n}{evaluate}}%
%BeginExpansion
n%
%EndExpansion
100,0.01\right) $. S\"{o}kt sannolikhet blir%
\begin{equation*}
P\left( X\leq 3\right) =\sum_{k=0}^{%
%TCIMACRO{\FORMULA{m}{m}{evaluate}}%
%BeginExpansion
m%
%EndExpansion
}\binom{%
%TCIMACRO{\FORMULA{n}{n}{evaluate}}%
%BeginExpansion
n%
%EndExpansion
}{k}%
%TCIMACRO{\FORMULA{p}{p}{evaluate}}%
%BeginExpansion
p%
%EndExpansion
^{k}\left( 1-%
%TCIMACRO{\FORMULA{p}{p}{evaluate}}%
%BeginExpansion
p%
%EndExpansion
\right) ^{%
%TCIMACRO{\FORMULA{n}{n}{evaluate}}%
%BeginExpansion
n%
%EndExpansion
-k}=%
%TCIMACRO{\FORMULA{\limfunc{svar}}{\limfunc{svar}}{evaluate}}%
%BeginExpansion
\limfunc{svar}%
%EndExpansion
\end{equation*}

\subsection{Variant}

\subsubsection{Setup}

$n:=\func{rand}(450,550)$

$m:=\func{rand}(5,15)$

$k:=\func{rand}(30,50)$

$\limfunc{svar}:=\limfunc{nplaces}\left( 1-\frac{\binom{k}{0}\binom{n-k}{m}}{%
\binom{n}{m}},2\right) $

\subsubsection{Statement}

F\"{o}r oml\"{a}ggning av ett tegeltak ink\"{o}ps ett parti med takpannor
som levereras i buntar om $100$ pannor i varje bunt. Man tar ut $%
%TCIMACRO{\FORMULA{m}{m}{evaluate}}%
%BeginExpansion
m%
%EndExpansion
$ buntar slumpm\"{a}ssigt och noterar antalet buntar som inneh\aa ller spr%
\"{a}ckta pannor. Om partiet best\aa r av $%
%TCIMACRO{\FORMULA{n}{n}{evaluate}}%
%BeginExpansion
n%
%EndExpansion
$ buntar och $%
%TCIMACRO{\FORMULA{k}{k}{evaluate}}%
%BeginExpansion
k%
%EndExpansion
$ av dessa inneh\aa ller spr\"{a}ckta pannor vad \"{a}r sannolikheten att n%
\aa gon av de studerade buntarna inneh\aa ller spr\"{a}ckta pannor?

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}$)

\paragraph{Solution}

S\"{a}tt%
\begin{equation*}
X=\text{antal buntar med spr\"{a}ckta pannor}
\end{equation*}%
det g\"{a}ller att $X\in Hyp\left( 
%TCIMACRO{\FORMULA{n}{n}{evaluate}}%
%BeginExpansion
n%
%EndExpansion
500,%
%TCIMACRO{\FORMULA{m}{m}{evaluate}}%
%BeginExpansion
m%
%EndExpansion
10,\frac{%
%TCIMACRO{\FORMULA{k}{k}{evaluate}}%
%BeginExpansion
k%
%EndExpansion
50}{%
%TCIMACRO{\FORMULA{n}{n}{evaluate}}%
%BeginExpansion
n%
%EndExpansion
500}\right) $. S\"{o}kt sannolikhet blir%
\begin{equation*}
P\left( X>0\right) =1-P\left( X=0\right) =1-\frac{\binom{%
%TCIMACRO{\FORMULA{k}{k}{evaluate}}%
%BeginExpansion
k%
%EndExpansion
}{0}\binom{%
%TCIMACRO{\FORMULA{n-k}{n-k}{evaluate}}%
%BeginExpansion
n-k%
%EndExpansion
}{%
%TCIMACRO{\FORMULA{m}{m}{evaluate}}%
%BeginExpansion
m%
%EndExpansion
}}{\binom{%
%TCIMACRO{\FORMULA{n}{n}{evaluate}}%
%BeginExpansion
n%
%EndExpansion
}{%
%TCIMACRO{\FORMULA{m}{m}{evaluate}}%
%BeginExpansion
m%
%EndExpansion
}}=%
%TCIMACRO{\FORMULA{\limfunc{svar}}{\limfunc{svar}}{evaluate}}%
%BeginExpansion
\limfunc{svar}%
%EndExpansion
\end{equation*}

\subsection{Variant}

\subsubsection{Setup}

$n:=\func{rand}(30,40)$

$m:=\func{rand}(10,15)$

$k:=\func{rand}(8,8)$

$l:=\left( \func{rand}(2,4),\left\lceil \frac{n}{7}\right\rceil \right) $

$q:=\limfunc{nplaces}\left( \frac{\binom{k}{l_{2}}\binom{n-k}{m-l_{2}}}{%
\binom{n}{m}},2\right) $

$I:=\left\{ 
\begin{array}{ccc}
1 & if & q\leq 0.05 \\ 
2 & if & q>0.05%
\end{array}%
\right. $

$\limfunc{svar}:=(\func{ja},\func{nej},\limfunc{nplaces}\left(
\sum_{x=0}^{l_{1}}\frac{\binom{k}{x}\binom{n-k}{m-x}}{\binom{n}{m}},2\right)
,q)$

\subsubsection{Statement}

En vetenskaplig expedition har inf\aa ngat och m\"{a}rkt $%
%TCIMACRO{\FORMULA{k}{k}{evaluate}}%
%BeginExpansion
k%
%EndExpansion
$ stycken sk\"{o}ldpaddor i ett visst omr\aa de. Expeditionen vet p\aa\ goda
grunder att det finns exakt $%
%TCIMACRO{\FORMULA{n}{n}{evaluate}}%
%BeginExpansion
n%
%EndExpansion
$ stycken sk\"{o}ldpaddor i omr\aa det varav $%
%TCIMACRO{\FORMULA{k}{k}{evaluate}}%
%BeginExpansion
k%
%EndExpansion
$ numera \"{a}r m\"{a}rkta. Efter en tid f\aa ngar expeditionen in $%
%TCIMACRO{\FORMULA{m}{m}{evaluate}}%
%BeginExpansion
m%
%EndExpansion
$ stycken sk\"{o}ldpaddor och noterar antalet m\"{a}rkta. Vad \"{a}r
sannolikheten f\"{o}r att det bland de $%
%TCIMACRO{\FORMULA{m}{m}{evaluate}}%
%BeginExpansion
m%
%EndExpansion
$ skall finnas h\"{o}gst $%
%TCIMACRO{\FORMULA{l_{1}}{l_{1}}{evaluate}}%
%BeginExpansion
l_{1}%
%EndExpansion
$ m\"{a}rkta om det finns $%
%TCIMACRO{\FORMULA{n}{n}{evaluate}}%
%BeginExpansion
n%
%EndExpansion
$ stycken totalt (svara med tv\aa\ decimaler)?

\subsubsection{Substatement}

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{3}$)

\subsubsection{Substatement}

Om det visar sig att det finns $%
%TCIMACRO{\FORMULA{l_{2}}{l_{2}}{evaluate}}%
%BeginExpansion
l_{2}%
%EndExpansion
$ m\"{a}rkta bland de $%
%TCIMACRO{\FORMULA{m}{m}{evaluate}}%
%BeginExpansion
m%
%EndExpansion
$ \"{a}r detta tillr\"{a}ckligt f\"{o}r att byta \textquotedblright goda
grunder\textquotedblright\ mot \textquotedblright l\"{o}sa
grunder\textquotedblright ? (Om sannolikheten f\"{o}r detta \"{a}r mindre 
\"{a}n $0.05$ svarar vi ja -- varf\"{o}r?)

\paragraph{Choices}

\begin{itemize}
\item $%
%TCIMACRO{%
%\FORMULA{\limfunc{svar}_{(I\func{mod}2)+1}}{\left( \limfunc{svar}\right) \left[ I+1\right] }{evaluate}}%
%BeginExpansion
\left( \limfunc{svar}\right) \left[ I+1\right] %
%EndExpansion
$

\item $%
%TCIMACRO{%
%\FORMULA{\limfunc{svar}_{I}}{\limfunc{svar}\left( I\right) }{evaluate}}%
%BeginExpansion
\limfunc{svar}\left( I\right) %
%EndExpansion
$\correctchoice{}
\end{itemize}

\paragraph{Solution}

S\"{a}tt%
\begin{equation*}
X=\text{antal m\"{a}rkta sk\"{o}ldpaddor i stickprovet}
\end{equation*}%
d\aa\ g\"{a}ller att $X\in Hyp\left( 
%TCIMACRO{\FORMULA{n}{n}{evaluate}}%
%BeginExpansion
n%
%EndExpansion
,%
%TCIMACRO{\FORMULA{m}{m}{evaluate}}%
%BeginExpansion
m%
%EndExpansion
,\frac{%
%TCIMACRO{\FORMULA{k}{k}{evaluate}}%
%BeginExpansion
k%
%EndExpansion
}{%
%TCIMACRO{\FORMULA{n}{n}{evaluate}}%
%BeginExpansion
n%
%EndExpansion
}\right) $. S\"{o}kt sannolikhet blir%
\begin{equation*}
P\left( X\leq 
%TCIMACRO{\FORMULA{l_{1}}{l_{1}}{evaluate}}%
%BeginExpansion
l_{1}%
%EndExpansion
\right) =\sum_{x=0}^{%
%TCIMACRO{\FORMULA{l_{1}}{l_{1}}{evaluate}}%
%BeginExpansion
l_{1}%
%EndExpansion
}\frac{\binom{%
%TCIMACRO{\FORMULA{k}{k}{evaluate}}%
%BeginExpansion
k%
%EndExpansion
}{x}\binom{%
%TCIMACRO{\FORMULA{n-k}{n-k}{evaluate}}%
%BeginExpansion
n-k%
%EndExpansion
}{%
%TCIMACRO{\FORMULA{m}{m}{evaluate}}%
%BeginExpansion
m%
%EndExpansion
-x}}{\binom{%
%TCIMACRO{\FORMULA{n}{n}{evaluate}}%
%BeginExpansion
n%
%EndExpansion
}{%
%TCIMACRO{\FORMULA{m}{m}{evaluate}}%
%BeginExpansion
m%
%EndExpansion
}}=%
%TCIMACRO{%
%\FORMULA{\limfunc{svar}_{3}}{\left( \limfunc{svar}\right) 3}{evaluate}}%
%BeginExpansion
\left( \limfunc{svar}\right) 3%
%EndExpansion
\end{equation*}%
Sannolikheten att finna $%
%TCIMACRO{\FORMULA{l_{2}}{l_{2}}{evaluate}}%
%BeginExpansion
l_{2}%
%EndExpansion
$ m\"{a}rkta \"{a}r%
\begin{equation*}
P\left( X=%
%TCIMACRO{\FORMULA{l_{2}}{l_{2}}{evaluate}}%
%BeginExpansion
l_{2}%
%EndExpansion
\right) =\frac{\binom{%
%TCIMACRO{\FORMULA{k}{k}{evaluate}}%
%BeginExpansion
k%
%EndExpansion
}{%
%TCIMACRO{\FORMULA{l_{2}}{l_{2}}{evaluate}}%
%BeginExpansion
l_{2}%
%EndExpansion
}\binom{%
%TCIMACRO{\FORMULA{n-k}{n-k}{evaluate}}%
%BeginExpansion
n-k%
%EndExpansion
}{%
%TCIMACRO{\FORMULA{m}{m}{evaluate}}%
%BeginExpansion
m%
%EndExpansion
-%
%TCIMACRO{\FORMULA{l_{2}}{l_{2}}{evaluate}}%
%BeginExpansion
l_{2}%
%EndExpansion
}}{\binom{%
%TCIMACRO{\FORMULA{n}{n}{evaluate}}%
%BeginExpansion
n%
%EndExpansion
}{%
%TCIMACRO{\FORMULA{m}{m}{evaluate}}%
%BeginExpansion
m%
%EndExpansion
}}=%
%TCIMACRO{\FORMULA{q}{q}{evaluate}}%
%BeginExpansion
q%
%EndExpansion
\text{.}
\end{equation*}%
Om denna sannolikhet \"{a}r mindre \"{a}n $0.05$ drar vi slutsatsen att
\textquotedblright goda grunder\textquotedblright\ \"{a}r \textquotedblright
l\"{o}sa grunder\textquotedblright .

\section{Question}

\subsection{Comment}

Negativ binomial-,Geometrisk- och Poissonf\"{o}rdelning

\subsection{Variant}

\subsubsection{Setup}

$p:=\func{rand}(\left\{ \frac{2}{3},\frac{3}{5},\frac{4}{5},\frac{5}{6}%
\right\} )$

$x:=\left( \func{rand}\left( 3,6\right) ,\func{rand}\left( 2,5\right)
\right) $

$\limfunc{svar}:=\left( \left\{ 1,2,3,\ldots \right\} ,p^{k-1}\left(
1-p\right) ,\limfunc{nplaces}\left( p^{x_{1}-1}\left( 1-p\right) ,3\right) ,%
\limfunc{nplaces}\left( \left( 1-p\right) \sum_{k=x_{2}}^{\infty
}p^{k-1},3\right) \right) $

\subsubsection{Statement}

I biljard forts\"{a}tter en spelare, enligt reglerna, att st\"{o}ta tills
han missar att f\aa\ ner en boll. Sannolikheten f\"{o}r en viss spelare att
missa \"{a}r $%
%TCIMACRO{\FORMULA{1-p}{1-p}{evaluate}}%
%BeginExpansion
1-p%
%EndExpansion
$. Om $X$ betecknar antalet st\"{o}tar spelaren g\"{o}r i en viss omg\aa ng
best\"{a}m:

\subsubsection{Substatement}

Utfallsrummet

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{1}$)

\subsubsection{Substatement}

Sannolikhetsfunktionen f\"{o}r $X$

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{2}$)

\subsubsection{Substatement}

Sannolikheten f\"{o}r exakt $%
%TCIMACRO{\FORMULA{x_{1}}{x_{1}}{evaluate}}%
%BeginExpansion
x_{1}%
%EndExpansion
$ st\"{o}tar

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{3}$)

\subsubsection{Substatement}

Sannolikheten f\"{o}r \aa tminstone $%
%TCIMACRO{\FORMULA{x_{2}}{x_{2}}{evaluate}}%
%BeginExpansion
x_{2}%
%EndExpansion
$ st\"{o}tar

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{4}$)

\paragraph{Solution}

F\"{o}ljande g\"{a}ller

\begin{enumerate}
\item $\Omega _{X}=\left\{ 1,2,3,\ldots \right\} 
%TCIMACRO{%
%\FORMULA{\limfunc{svar}_{1}}{\left( \limfunc{svar}\right) 1}{evaluate}}%
%BeginExpansion
\left( \limfunc{svar}\right) 1%
%EndExpansion
$

\item $P\left( X=k\right) =\left( 
%TCIMACRO{\FORMULA{1-p}{1-p}{evaluate}}%
%BeginExpansion
1-p%
%EndExpansion
\right) 
%TCIMACRO{\FORMULA{p}{p}{evaluate}}%
%BeginExpansion
p%
%EndExpansion
^{k-1}\quad k\in \Omega _{X}%
%TCIMACRO{%
%\FORMULA{\limfunc{svar}_{2}}{\left( \limfunc{svar}\right) 2}{evaluate}}%
%BeginExpansion
\left( \limfunc{svar}\right) 2%
%EndExpansion
$

\item $P\left( X=%
%TCIMACRO{\FORMULA{x_{1}}{x_{1}}{evaluate}}%
%BeginExpansion
x_{1}%
%EndExpansion
5\right) =\left( 
%TCIMACRO{\FORMULA{1-p}{1-p}{evaluate}}%
%BeginExpansion
1-p%
%EndExpansion
\right) 
%TCIMACRO{\FORMULA{p}{p}{evaluate}}%
%BeginExpansion
p%
%EndExpansion
^{%
%TCIMACRO{\FORMULA{x_{1}}{x_{1}}{evaluate}}%
%BeginExpansion
x_{1}%
%EndExpansion
-1}=%
%TCIMACRO{%
%\FORMULA{\limfunc{svar}_{3}}{\left( \limfunc{svar}\right) 3}{evaluate}}%
%BeginExpansion
\left( \limfunc{svar}\right) 3%
%EndExpansion
$

\item $P\left( X\geq 
%TCIMACRO{\FORMULA{x_{2}}{x_{2}}{evaluate}}%
%BeginExpansion
x_{2}%
%EndExpansion
4\right) =\sum_{k=%
%TCIMACRO{\FORMULA{x_{2}}{x_{2}}{evaluate}}%
%BeginExpansion
x_{2}%
%EndExpansion
}^{\infty }\left( 
%TCIMACRO{\FORMULA{1-p}{1-p}{evaluate}}%
%BeginExpansion
1-p%
%EndExpansion
\right) 
%TCIMACRO{\FORMULA{p}{p}{evaluate}}%
%BeginExpansion
p%
%EndExpansion
^{k-1}=%
%TCIMACRO{%
%\FORMULA{\limfunc{svar}_{4}}{\left( \limfunc{svar}\right) 4}{evaluate}}%
%BeginExpansion
\left( \limfunc{svar}\right) 4%
%EndExpansion
$
\end{enumerate}

\subsection{Variant}

\subsubsection{Setup}

$k:=\func{rand}(3,6)$

$l:=\func{rand}\left( 3,k\right) $

$p:=\limfunc{nplaces}\left( \func{rand}(4,6)/10,1\right) $

$\limfunc{svar}:=\limfunc{nplaces}\left( \binom{k-1}{l-1}p^{l}\left(
1-p\right) ^{k-l},3\right) $

\subsubsection{Statement}

Sannolikheten att f\aa\ en pojke \"{a}r $%
%TCIMACRO{\FORMULA{p}{p}{evaluate}}%
%BeginExpansion
p%
%EndExpansion
$. Ett par har best\"{a}mt sig f\"{o}r att ha exakt $%
%TCIMACRO{\FORMULA{l}{l}{evaluate}}%
%BeginExpansion
l%
%EndExpansion
$ pojkar. Vad \"{a}r sannolikheten f\"{o}r att familjen kommer best\aa\ av $%
%TCIMACRO{\FORMULA{l}{l}{evaluate}}%
%BeginExpansion
l%
%EndExpansion
$ pojkar och $%
%TCIMACRO{\FORMULA{k-l}{k-l}{evaluate}}%
%BeginExpansion
k-l%
%EndExpansion
$ flickor/a? (svara med tre decimaler)

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}$)

\paragraph{Solution}

S\"{a}tt%
\begin{equation*}
X_{i}=\text{antal barn tills }i\text{:te pojken\quad }i=1,2,\ldots ,%
%TCIMACRO{\FORMULA{l}{l}{evaluate}}%
%BeginExpansion
l%
%EndExpansion
\end{equation*}%
det g\"{a}ller d\aa\ att $X_{i}\in Geom\left( 
%TCIMACRO{\FORMULA{p}{p}{evaluate}}%
%BeginExpansion
p%
%EndExpansion
\right) $. Det totala antalet barn, n\"{a}r vi har exakt $%
%TCIMACRO{\FORMULA{l}{l}{evaluate}}%
%BeginExpansion
l%
%EndExpansion
$ pojkar, blir d\aa 
\begin{equation*}
X=\sum_{i=1}^{%
%TCIMACRO{\FORMULA{l}{l}{evaluate}}%
%BeginExpansion
l%
%EndExpansion
}X_{i}\in NegBin\left( 
%TCIMACRO{\FORMULA{l}{l}{evaluate}}%
%BeginExpansion
l%
%EndExpansion
,%
%TCIMACRO{\FORMULA{p}{p}{evaluate}}%
%BeginExpansion
p%
%EndExpansion
\right) \text{.}
\end{equation*}%
H\"{a}rav f\"{o}ljer%
\begin{equation*}
P\left( X=%
%TCIMACRO{\FORMULA{k}{k}{evaluate}}%
%BeginExpansion
k%
%EndExpansion
\right) =\binom{%
%TCIMACRO{\FORMULA{k}{k}{evaluate}}%
%BeginExpansion
k%
%EndExpansion
-1}{%
%TCIMACRO{\FORMULA{l}{l}{evaluate}}%
%BeginExpansion
l%
%EndExpansion
-1}%
%TCIMACRO{\FORMULA{p}{p}{evaluate}}%
%BeginExpansion
p%
%EndExpansion
^{%
%TCIMACRO{\FORMULA{l}{l}{evaluate}}%
%BeginExpansion
l%
%EndExpansion
}\left( 1-%
%TCIMACRO{\FORMULA{p}{p}{evaluate}}%
%BeginExpansion
p%
%EndExpansion
\right) ^{%
%TCIMACRO{\FORMULA{k}{k}{evaluate}}%
%BeginExpansion
k%
%EndExpansion
-%
%TCIMACRO{\FORMULA{l}{l}{evaluate}}%
%BeginExpansion
l%
%EndExpansion
}=%
%TCIMACRO{\FORMULA{\limfunc{svar}}{\limfunc{svar}}{evaluate}}%
%BeginExpansion
\limfunc{svar}%
%EndExpansion
\text{.}
\end{equation*}

\subsection{Variant}

\subsubsection{Setup}

$\lambda :=\func{rand}(4,14)$

$k:=\func{rand}\left( \left\lfloor \frac{\lambda }{2}\right\rfloor ,2\lambda
\right) $

$m:=\func{rand}\left( \lambda ,\lambda +4\right) $

$\limfunc{svar}:=\left( \limfunc{nplaces}\left( \frac{\lambda ^{0}}{0!}%
e^{-\lambda },3\right) ,\limfunc{nplaces}\left( e^{-\lambda }\sum_{x=0}^{k}%
\frac{\lambda ^{x}}{x!},3\right) ,\limfunc{nplaces}\left( e^{-\lambda
}\sum_{x=m+1}^{\infty }\frac{\lambda ^{x}}{x!},3\right) \right) $

\subsubsection{Statement}

Antalet beg\"{a}rda k\"{o}rningar, per minut, p\aa\ en CRAYdator f\"{o}ljer
en Poissonf\"{o}rdelning med $\lambda =%
%TCIMACRO{\FORMULA{\lambda }{\lambda }{evaluate}}%
%BeginExpansion
\lambda %
%EndExpansion
$ (svara med tre decimaler).

\subsubsection{Substatement}

Best\"{a}m sannolikheten att ingen k\"{o}rning har beg\"{a}rts f\"{o}r en
given minut.

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{1}$)

\subsubsection{Substatement}

Best\"{a}m sannolikheten att $%
%TCIMACRO{\FORMULA{k}{k}{evaluate}}%
%BeginExpansion
k%
%EndExpansion
$\ eller f\"{a}rre k\"{o}rningar beg\"{a}rts f\"{o}r en given minut.

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{2}$)

\subsubsection{Substatement}

Om mer \"{a}n $%
%TCIMACRO{\FORMULA{m}{m}{evaluate}}%
%BeginExpansion
m%
%EndExpansion
$ k\"{o}rningar beg\"{a}rts f\"{o}r en given minut uppst\aa r v\"{a}ntetid.
Vad \"{a}r sannolikheten f\"{o}r detta?

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{3}$)

\paragraph{Solution}

S\"{a}tt%
\begin{equation*}
X=\text{antal beg\"{a}rda k\"{o}rningar per minut}
\end{equation*}%
det g\"{a}ller att $X\in Po\left( 
%TCIMACRO{\FORMULA{\lambda }{\lambda }{evaluate}}%
%BeginExpansion
\lambda %
%EndExpansion
\right) $. S\"{o}kta sannolikheter blir

\begin{enumerate}
\item $P\left( X=0\right) =\frac{%
%TCIMACRO{\FORMULA{\lambda }{\lambda }{evaluate}}%
%BeginExpansion
\lambda %
%EndExpansion
^{0}}{0!}e^{-%
%TCIMACRO{\FORMULA{\lambda }{\lambda }{evaluate}}%
%BeginExpansion
\lambda %
%EndExpansion
}=%
%TCIMACRO{%
%\FORMULA{\limfunc{svar}_{1}}{\left( \limfunc{svar}\right) 1}{evaluate}}%
%BeginExpansion
\left( \limfunc{svar}\right) 1%
%EndExpansion
$

\item $P\left( X\leq 2\right) =e^{-%
%TCIMACRO{\FORMULA{\lambda }{\lambda }{evaluate}}%
%BeginExpansion
\lambda %
%EndExpansion
}\sum_{x=0}^{%
%TCIMACRO{\FORMULA{k}{k}{evaluate}}%
%BeginExpansion
k%
%EndExpansion
}\frac{%
%TCIMACRO{\FORMULA{\lambda }{\lambda }{evaluate}}%
%BeginExpansion
\lambda %
%EndExpansion
^{x}}{x!}=%
%TCIMACRO{%
%\FORMULA{\limfunc{svar}_{2}}{\left( \limfunc{svar}\right) 2}{evaluate}}%
%BeginExpansion
\left( \limfunc{svar}\right) 2%
%EndExpansion
$

\item $P\left( X>10\right) =1-e^{-%
%TCIMACRO{\FORMULA{\lambda }{\lambda }{evaluate}}%
%BeginExpansion
\lambda %
%EndExpansion
}\sum_{x=0}^{%
%TCIMACRO{\FORMULA{m}{m}{evaluate}}%
%BeginExpansion
m%
%EndExpansion
}\frac{%
%TCIMACRO{\FORMULA{\lambda }{\lambda }{evaluate}}%
%BeginExpansion
\lambda %
%EndExpansion
^{x}}{x!}=%
%TCIMACRO{%
%\FORMULA{\limfunc{svar}_{3}}{\left( \limfunc{svar}\right) 3}{evaluate}}%
%BeginExpansion
\left( \limfunc{svar}\right) 3%
%EndExpansion
$
\end{enumerate}

\subsection{Variant}

\subsubsection{Setup}

$\lambda :=\func{rand}(2,8)$

$k:=\func{rand}\left( 0,2\lambda \right) $

$m:=\func{rand}\left( k+2,k+\lambda \right) $

$\limfunc{svar}:=\limfunc{nplaces}\left( e^{-\lambda }\sum_{x=k+1}^{m-1}%
\frac{\lambda ^{x}}{x!},3\right) $

\subsubsection{Statement}

Antalet kunder som i ett visst tidsintervall anl\"{a}nder till ett betj\"{a}%
ningsst\"{a}lle kan ofta approximativt beskrivas med en Poissonf\"{o}rdelad
stokastisk variabel $X\,$\ med parametern $\lambda $. Om $\lambda =%
%TCIMACRO{\FORMULA{\lambda }{\lambda }{evaluate}}%
%BeginExpansion
\lambda %
%EndExpansion
$ vad \"{a}r sannolikheten att fler \"{a}n $%
%TCIMACRO{\FORMULA{k}{k}{evaluate}}%
%BeginExpansion
k%
%EndExpansion
$ men f\"{a}rre \"{a}n $%
%TCIMACRO{\FORMULA{m}{m}{evaluate}}%
%BeginExpansion
m%
%EndExpansion
$ kunder anl\"{a}nder under tidsintervallet ifr\aa ga?

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}$)

\paragraph{Solution}

S\"{a}tt%
\begin{equation*}
X=\text{antalet kunder under ett tidsintervall}
\end{equation*}%
der g\"{a}ller att $X\in Po\left( 
%TCIMACRO{\FORMULA{\lambda }{\lambda }{evaluate}}%
%BeginExpansion
\lambda %
%EndExpansion
\right) $. S\"{o}kt sannolikhet erh\aa lls nu till 
\begin{equation*}
P\left( 
%TCIMACRO{\FORMULA{k}{k}{evaluate}}%
%BeginExpansion
k%
%EndExpansion
<X<%
%TCIMACRO{\FORMULA{m}{m}{evaluate}}%
%BeginExpansion
m%
%EndExpansion
\right) =\sum_{x=%
%TCIMACRO{\FORMULA{k}{k}{evaluate}}%
%BeginExpansion
k%
%EndExpansion
+1}^{%
%TCIMACRO{\FORMULA{m}{m}{evaluate}}%
%BeginExpansion
m%
%EndExpansion
-1}\frac{%
%TCIMACRO{\FORMULA{\lambda }{\lambda }{evaluate}}%
%BeginExpansion
\lambda %
%EndExpansion
^{x}}{x!}e^{-%
%TCIMACRO{\FORMULA{\lambda }{\lambda }{evaluate}}%
%BeginExpansion
\lambda %
%EndExpansion
}=%
%TCIMACRO{\FORMULA{\limfunc{svar}}{\limfunc{svar}}{evaluate}}%
%BeginExpansion
\limfunc{svar}%
%EndExpansion
\end{equation*}

\section{Question}

\subsection{Comment}

Allm\"{a}na kontinuerliga f\"{o}rdelningar och den likformiga f\"{o}%
rdelningen

\subsection{Variant}

\subsubsection{Setup}

$k:=\func{rand}(\left\{ \frac{3}{4},\frac{2}{9}\right\} )$

$a:=\left( \frac{6.0}{k}\right) ^{\frac{1}{3}}$

$\limfunc{svar}:=\left( a,\limfunc{nplaces}\left( \int_{1}^{a}kx\left(
a-x\right) \,dx,2\right) \right) $

\subsubsection{Statement}

Funktionen $f\left( x\right) $ \"{a}r definierad enligt 
\begin{equation*}
f\left( x\right) =\left\{ 
\begin{array}{ll}
%TCIMACRO{\FORMULA{k}{k}{evaluate}}%
%BeginExpansion
k%
%EndExpansion
x\left( a-x\right) & \text{om }0\leq x\leq a \\ 
0 & \text{annars}%
\end{array}%
\right. \text{.}
\end{equation*}

\subsubsection{Substatement}

Best\"{a}m $a$ s\aa\ att $f\left( x\right) $ blir en t\"{a}thetsfunktion.

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{1}$)

\subsubsection{Substatement}

Ber\"{a}kna d\"{a}refter sannolikheten att $X$ ligger i intervallet $\left[
1,3\right] $.

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{2}$)

\subsubsection{Solution}

Eftersom $f\left( x\right) \geq 0$ f\"{o}r alla $x$, oavsett v\"{a}rdet p%
\aa\ $a$, s\aa\ best\"{a}ms $a$ av 
\begin{equation*}
1=\int_{-\infty }^{\infty }f\left( x\right) \,dx=\int_{0}^{%
%TCIMACRO{\FORMULA{a}{a}{evaluate}}%
%BeginExpansion
a%
%EndExpansion
}%
%TCIMACRO{\FORMULA{k}{k}{evaluate}}%
%BeginExpansion
k%
%EndExpansion
x\left( 
%TCIMACRO{\FORMULA{a}{a}{evaluate}}%
%BeginExpansion
a%
%EndExpansion
-x\right) \,dx=%
%TCIMACRO{\FORMULA{k}{k}{evaluate}}%
%BeginExpansion
k%
%EndExpansion
\left[ a\frac{x^{2}}{2}-\frac{x^{3}}{3}\right] _{0}^{a}=%
%TCIMACRO{\FORMULA{k}{k}{evaluate}}%
%BeginExpansion
k%
%EndExpansion
\frac{a^{3}}{6}
\end{equation*}%
vilket ger $a=\left( \frac{6}{%
%TCIMACRO{\FORMULA{k}{k}{evaluate}}%
%BeginExpansion
k%
%EndExpansion
}\right) ^{\frac{1}{3}}=%
%TCIMACRO{%
%\FORMULA{\limfunc{svar}_{1}}{\left( \limfunc{svar}\right) 1}{evaluate}}%
%BeginExpansion
\left( \limfunc{svar}\right) 1%
%EndExpansion
$.

F\"{o}r sannolikheten finner vi 
\begin{equation*}
P\left( 1\leq X\leq 3\right) =\int_{1}^{%
%TCIMACRO{\FORMULA{a}{a}{evaluate}}%
%BeginExpansion
a%
%EndExpansion
}%
%TCIMACRO{\FORMULA{k}{k}{evaluate}}%
%BeginExpansion
k%
%EndExpansion
x\left( 2-x\right) \,dx=%
%TCIMACRO{\FORMULA{k}{k}{evaluate}}%
%BeginExpansion
k%
%EndExpansion
\left[ 
%TCIMACRO{\FORMULA{a}{a}{evaluate}}%
%BeginExpansion
a%
%EndExpansion
\frac{x^{2}}{2}-\frac{x^{3}}{3}\right] _{1}^{2}=%
%TCIMACRO{%
%\FORMULA{\limfunc{svar}_{2}}{\left( \limfunc{svar}\right) 2}{evaluate}}%
%BeginExpansion
\left( \limfunc{svar}\right) 2%
%EndExpansion
\end{equation*}

\subsection{Variant}

\subsubsection{Statement}

Den stokastiska variabeln $X$ har f\"{o}rdelningsfunktionen%
\begin{equation*}
F\left( x\right) =\left\{ 
\begin{array}{ccc}
0 & \text{om} & x\leq-1 \\ 
\frac{1}{4}\left( 1+x\right) ^{2} & \text{om} & -1<x\leq1 \\ 
1 & \text{om} & 1<x%
\end{array}%
\right.
\end{equation*}
Best\"{a}m t\"{a}thetsfunktionen f\"{o}r $Y=|X|$.

\paragraph{Choices}

\begin{itemize}
\item Likformig f\"{o}rdelning\correctchoice{}

\item Binomial-f\"{o}rdelning

\item Beta-f\"{o}rdelning

\item Normal-f\"{o}rdelning

\item Kvadratisk f\"{o}rdelning

\item Exponentiel f\"{o}rdelning

\item Annan f\"{o}rdelning
\end{itemize}

\subsubsection{Solution}

F\"{o}r $y>0$ erh\aa lls%
\begin{align*}
F_{Y}\left( y\right) & =P\left( Y\leq y\right) \\
& =P\left( |X|\leq y\right) \\
& =P\left( -y\leq X\leq y\right) \\
\left\{ \text{ty }X\text{ kontinuerlig}\right\} & =P\left( X\leq y\right)
-P\left( X\leq -y\right) \\
\left\{ \text{om }y\leq 1\right\} & =\frac{1}{4}\left( 1+y\right) ^{2}-\frac{%
1}{4}\left( 1-y\right) ^{2}=y \\
\left\{ \text{om }y>1\right\} & =1-0=1
\end{align*}%
varav%
\begin{equation*}
F_{Y}\left( y\right) =\left\{ 
\begin{array}{ccc}
0 & \text{om} & y\leq 0 \\ 
y & \text{om} & 0<y\leq 1 \\ 
1 & \text{om} & 1<y%
\end{array}%
\right.
\end{equation*}%
och derivering ger%
\begin{equation*}
f_{Y}\left( y\right) =\left\{ 
\begin{array}{ccc}
y & \text{om} & 0<y\leq 1 \\ 
0 & \text{om} & \text{annars}%
\end{array}%
\right.
\end{equation*}

\subsection{Variant}

\subsubsection{Setup}

$\lambda :=\func{rand}(2,5)$

\subsubsection{Statement}

Den stokastiska variabeln $X$ har t\"{a}thetsfunktionen%
\begin{equation*}
f\left( x\right) =\left\{ 
\begin{array}{lcl}
%TCIMACRO{\FORMULA{\lambda }{\lambda }{evaluate}}%
%BeginExpansion
\lambda %
%EndExpansion
x^{2} & \text{om} & 0<x<1 \\ 
0 & \text{om} & \text{annars}%
\end{array}%
\right.
\end{equation*}%
Best\"{a}m t\"{a}thetsfunktionen f\"{o}r den stokastiska variabeln $Y=\ln X$.

\paragraph{Choices}

\begin{itemize}
\item Likformig f\"{o}rdelning

\item Binomial-f\"{o}rdelning

\item Beta-f\"{o}rdelning

\item Normal-f\"{o}rdelning

\item Kvadratisk f\"{o}rdelning

\item Exponential-f\"{o}rdelning

\item Annan f\"{o}rdelning\correctchoice{}
\end{itemize}

\subsubsection{Solution}

Utfallsrummet f\"{o}r $Y$ blir $\Omega _{Y}=\left\{ y\in R\mid -\infty
<y<0\right\} $ och f\"{o}ljande likheter erh\aa lls%
\begin{align*}
F_{Y}\left( y\right) & =P\left( Y\leq y\right) =P\left( \ln X\leq y\right) \\
& =P\left( X\leq e^{y}\right) =\int_{0}^{e^{y}}%
%TCIMACRO{\FORMULA{\lambda }{\lambda }{evaluate}}%
%BeginExpansion
\lambda %
%EndExpansion
x^{2}\,dx \\
& =e^{%
%TCIMACRO{\FORMULA{\lambda }{\lambda }{evaluate}}%
%BeginExpansion
\lambda %
%EndExpansion
y}
\end{align*}

t\"{a}thetsfunktionen \"{a}r d\"{a}rf\"{o}r%
\begin{equation*}
f_{Y}\left( y\right) =\left\{ 
\begin{array}{lcl}
e^{%
%TCIMACRO{\FORMULA{\lambda }{\lambda }{evaluate}}%
%BeginExpansion
\lambda %
%EndExpansion
y} & \text{om} & y<0 \\ 
0 & \text{om} & \text{annars}%
\end{array}%
\right.
\end{equation*}

\subsection{Variant}

\subsubsection{Setup}

$a:=\func{rand}(0,3)$

$b:=\func{rand}(a+3,a+7)$

$l:=\func{rand}\left( a+1,b-1\right) $

$\limfunc{svar}:=\limfunc{nplaces}\left( \frac{l-a}{b-a},2\right) $

\subsubsection{Statement}

Ett reellt tal tas p\aa\ m\aa f\aa\ mellan $%
%TCIMACRO{\FORMULA{a}{a}{evaluate}}%
%BeginExpansion
a%
%EndExpansion
$ och $%
%TCIMACRO{\FORMULA{b}{b}{evaluate}}%
%BeginExpansion
b%
%EndExpansion
$. Detta tal best\"{a}mmer sidl\"{a}ngden p\aa\ en kvadrat. Best\"{a}m
sannolikheten f\"{o}r att denna kvadrats area \"{a}r h\"{o}gst $%
%TCIMACRO{\FORMULA{l^{2}}{l^{2}}{evaluate}}%
%BeginExpansion
l^{2}%
%EndExpansion
$ (svara med tv\aa\ decimaler).

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}$)

\subsubsection{Solution}

S\"{a}tt%
\begin{equation*}
X=\text{kvadratens sidl\"{a}ngd}
\end{equation*}%
d\aa\ g\"{a}ller att $X\in R\left( 
%TCIMACRO{\FORMULA{a}{a}{evaluate}}%
%BeginExpansion
a%
%EndExpansion
,%
%TCIMACRO{\FORMULA{b}{b}{evaluate}}%
%BeginExpansion
b%
%EndExpansion
\right) $. S\"{o}kt sannolikhet kan nu skrivas%
\begin{align*}
P\left( X^{2}\leq 
%TCIMACRO{\FORMULA{l^{2}}{l^{2}}{evaluate}}%
%BeginExpansion
l^{2}%
%EndExpansion
\right) & =P\left( X\leq 
%TCIMACRO{\FORMULA{l}{l}{evaluate}}%
%BeginExpansion
l%
%EndExpansion
\right) =\frac{%
%TCIMACRO{\FORMULA{l}{l}{evaluate}}%
%BeginExpansion
l%
%EndExpansion
-%
%TCIMACRO{\FORMULA{a}{a}{evaluate}}%
%BeginExpansion
a%
%EndExpansion
}{%
%TCIMACRO{\FORMULA{b}{b}{evaluate}}%
%BeginExpansion
b%
%EndExpansion
-%
%TCIMACRO{\FORMULA{a}{a}{evaluate}}%
%BeginExpansion
a%
%EndExpansion
} \\
& =%
%TCIMACRO{\FORMULA{\limfunc{svar}}{\limfunc{svar}}{evaluate}}%
%BeginExpansion
\limfunc{svar}%
%EndExpansion
\end{align*}

\subsection{Variant}

\subsubsection{Setup}

$\limfunc{svar}:=\left( \left\{ 20,25,30\right\} ,\left\{ 0,\frac{1}{4},%
\frac{5}{8},1\right\} \right) $

\subsubsection{Statement}

Avgiften i ett parkeringshus best\aa r av en fast avgift om $10$ kronor samt
en r\"{o}rlig avgift om $5$ kronor f\"{o}r varje p\aa b\"{o}rjad $15$%
-minuters period. N\"{a}r fru Svensson parkerar sin bil f\"{o}r att handla i
ett n\"{a}rbel\"{a}get varuhus kan man anse att hennes parkeringstid \"{a}r
likformigt f\"{o}rdelad p\aa\ intervallet $20$ till $60$ minuter. Best\"{a}m
sannolikhetsfunktionen f\"{o}r den avgift som hon f\aa r betala.

\subsubsection{Substatement}

Ange de utfallsrummet f\"{o}r kostnaden (svara p\aa\ formen $\left\{
a_{1},a_{2},a_{3},\ldots \right\} $)

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{1}$)

\subsubsection{Substatement}

Ange kostnadens f\"{o}rdelningsfunktion (svara p\aa\ formen $\left\{ F\left(
a_{1}\right) ,F\left( a_{2}\right) ,F\left( a_{3}\right) ,\ldots \right\} $).

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{2}$)

\subsubsection{Solution}

S\"{a}tt 
\begin{align*}
X& =\text{parkeringstiden} \\
Y& =\text{parkeringsavgiften}
\end{align*}%
d\aa\ g\"{a}ller att $X\in R\left( 20,60\right) $ samt att%
\begin{equation*}
Y=10+\left\{ 
\begin{array}{ccc}
10 & \text{om} & 20\leq X<30 \\ 
15 & \text{om} & 30\leq X<45 \\ 
20 & \text{om} & 45\leq X<60%
\end{array}%
\right.
\end{equation*}%
Sannolikhetsfunktionen erh\aa lls nu till 
\begin{align*}
P\left( Y=20\right) & =P\left( 20\leq X<30\right) =\frac{30-20}{60-20} \\
& =\frac{1}{4} \\
P\left( Y=25\right) & =P\left( 30\leq X<45\right) =\frac{45-30}{60-20} \\
& =\frac{3}{8} \\
P\left( Y=30\right) & =P\left( 45\leq X<60\right) =\frac{60-45}{60-20} \\
& =\frac{3}{8}
\end{align*}

\section{Question}

\subsection{Comment}

Exponential-, normal-, gamma- och Weibullf\"{o}rdelning

\subsection{Variant}

\subsubsection{Setup}

$a:=\func{rand}(1,4)$

$b:=\func{rand}(a+2,10)$

$\lambda :=\func{rand}\left( \left\{ 0.5,0.6,0.7,1,1.2\right\} \right) $

$\limfunc{svar}:=\left( 1-e^{-\lambda t},\limfunc{nplaces}\left(
\int_{a}^{b}\lambda e^{-\lambda t}dt,2\right) \right) $

\subsubsection{Statement}

L\"{a}ngden av ett telefonsamtal kan ofta beskrivas p\aa\ f\"{o}ljande s\"{a}%
tt: Sannolikheten att ett samtal varar l\"{a}ngre \"{a}n $t$ minuter \"{a}r $%
e^{-\lambda t}$ d\"{a}r $\lambda $ \"{a}r en positiv konstant. Definiera den
stokastiska variabeln%
\begin{equation*}
X=\text{ tiden f\"{o}r ett samtal.}
\end{equation*}

\subsubsection{Substatement}

Best\"{a}m f\"{o}rdelningsfunktionen f\"{o}r $X$ (svara p\aa\ formen $%
F\left( t\right) $).

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{1}$)

\subsubsection{Substatement}

Ber\"{a}kna $P\left( 
%TCIMACRO{\FORMULA{a}{a}{evaluate}}%
%BeginExpansion
a%
%EndExpansion
<X\leq 
%TCIMACRO{\FORMULA{b}{b}{evaluate}}%
%BeginExpansion
b%
%EndExpansion
\right) $ n\"{a}r $\lambda =%
%TCIMACRO{\FORMULA{\lambda }{\lambda }{evaluate}}%
%BeginExpansion
\lambda %
%EndExpansion
$ (svara med tv\aa\ decimaler).

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{2}$)

\subsubsection{Solution}

H\"{a}ndelsen \textquotedblright ett samtal varar l\"{a}ngre \"{a}n $t$
minuter\textquotedblright\ kan skrivas $X>t$ varf\"{o}r det g\"{a}ller%
\begin{equation*}
P\left( X>t\right) =e^{-%
%TCIMACRO{\FORMULA{\lambda }{\lambda }{evaluate}}%
%BeginExpansion
\lambda %
%EndExpansion
t}
\end{equation*}%
vilket ger f\"{o}rdelningsfunktionen%
\begin{equation*}
F\left( x\right) =1-e^{-%
%TCIMACRO{\FORMULA{\lambda }{\lambda }{evaluate}}%
%BeginExpansion
\lambda %
%EndExpansion
t}
\end{equation*}%
samt s\"{o}kt sannolikhet%
\begin{align*}
P\left( 
%TCIMACRO{\FORMULA{a}{a}{evaluate}}%
%BeginExpansion
a%
%EndExpansion
<X\leq 
%TCIMACRO{\FORMULA{b}{b}{evaluate}}%
%BeginExpansion
b%
%EndExpansion
\right) & =F\left( 
%TCIMACRO{\FORMULA{b}{b}{evaluate}}%
%BeginExpansion
b%
%EndExpansion
\right) -F\left( 
%TCIMACRO{\FORMULA{a}{a}{evaluate}}%
%BeginExpansion
a%
%EndExpansion
\right) =1-e^{-%
%TCIMACRO{\FORMULA{b}{b}{evaluate}}%
%BeginExpansion
b%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{\lambda }{\lambda }{evaluate}}%
%BeginExpansion
\lambda %
%EndExpansion
}-\left( 1-e^{-%
%TCIMACRO{\FORMULA{a}{a}{evaluate}}%
%BeginExpansion
a%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{\lambda }{\lambda }{evaluate}}%
%BeginExpansion
\lambda %
%EndExpansion
}\right) \\
& =%
%TCIMACRO{%
%\FORMULA{\limfunc{svar}_{2}}{\left( \limfunc{svar}\right) 2}{evaluate}}%
%BeginExpansion
\left( \limfunc{svar}\right) 2%
%EndExpansion
\end{align*}

\subsection{Variant}

\subsubsection{Setup}

$\lambda :=\limfunc{nplaces}\left( \func{rand}(50,400)/100,2\right) $

$\limfunc{svar}:=\limfunc{nplaces}\left( 0.36788,3\right) $

\subsubsection{Statement}

Avst\aa ndet mellan bilar i gles trafik kan ofta beskrivas med en
exponentialf\"{o}rdelad variabel med parametern $%
%TCIMACRO{\FORMULA{\lambda }{\lambda }{evaluate}}%
%BeginExpansion
\lambda %
%EndExpansion
$. Best\"{a}m sannolikheten f\"{o}r att avst\aa ndet \"{a}r st\"{o}rre \"{a}%
n $\frac{2}{%
%TCIMACRO{\FORMULA{\lambda }{\lambda }{evaluate}}%
%BeginExpansion
\lambda %
%EndExpansion
}$ givet att det \"{a}r st\"{o}rre \"{a}n $\frac{1}{%
%TCIMACRO{\FORMULA{\lambda }{\lambda }{evaluate}}%
%BeginExpansion
\lambda %
%EndExpansion
}$.

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}$)

\subsubsection{Solution}

S\"{a}tt%
\begin{equation*}
X=\text{avst\aa nd mellan tv\aa\ bilar i gles trafik}
\end{equation*}%
d\aa\ g\"{a}ller att $X\in Exp\left( \lambda \right) $. S\"{o}kes%
\begin{align*}
P\left( X>\frac{2}{\lambda }\mid X>\frac{1}{\lambda }\right) & =\frac{%
P\left( X>\frac{2}{\lambda },X>\frac{1}{\lambda }\right) }{P\left( X>\frac{1%
}{\lambda }\right) } \\
& =\frac{P\left( X>\frac{2}{\lambda }\right) }{P\left( X>\frac{1}{\lambda }%
\right) }=\frac{1-\left( 1-e^{-\lambda \frac{2}{\lambda }}\right) }{1-\left(
1-e^{-\lambda \frac{1}{\lambda }}\right) } \\
& =\frac{e^{-2}}{e^{-1}} \\
& =e^{-1}\text{.}
\end{align*}%
Resultatet \"{a}r d\"{a}rf\"{o}r oberoende av v\"{a}rdet p\aa\ parametern $%
\lambda $.

\subsection{Variant}

\subsubsection{Setup}

$\mu :=\func{rand}(1,6)$

$\sigma :=\func{rand}(\left\{ 0.5,1,1.5,2,2.5,3\right\} )$

$a:=6$

$b:=1.9$

$c:=5.6$

$\alpha :=0.05$

$\limfunc{svar}:=\left( \limfunc{nplaces}\left( \func{NormalDist}\left(
a;\mu ,\sigma \right) ,2\right) ,\limfunc{nplaces}\left( \func{NormalDist}%
\left( c;\mu ,\sigma \right) -\func{NormalDist}\left( b;\mu ,\sigma \right)
,2\right) ,\limfunc{nplaces}\left( \func{NormalInv}\left( 1-\alpha ;\mu
,\sigma \right) ,2\right) ,\limfunc{nplaces}\left( \func{NormalInv}\left( 1-%
\frac{\alpha }{2};\mu ,\sigma \right) ,2\right) \right) $

\subsubsection{Statement}

Den stokastiska variabeln $X$ \"{a}r normalf\"{o}rdelad med $\mu =4$ och $%
\sigma =2$. Best\"{a}m

\subsubsection{Substatement}

$P\left( X\leq 
%TCIMACRO{\FORMULA{a}{a}{evaluate}}%
%BeginExpansion
a%
%EndExpansion
\right) =$

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{1}$)

\subsubsection{Substatement}

$P\left( 
%TCIMACRO{\FORMULA{b}{b}{evaluate}}%
%BeginExpansion
b%
%EndExpansion
<X<%
%TCIMACRO{\FORMULA{c}{c}{evaluate}}%
%BeginExpansion
c%
%EndExpansion
\right) =$

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{2}$)

\subsubsection{Substatement}

Det g\"{a}ller att $P\left( X\leq x\right) =%
%TCIMACRO{\FORMULA{1-\alpha }{1-\alpha }{evaluate}}%
%BeginExpansion
1-\alpha %
%EndExpansion
$. Best\"{a}m $x=$

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{3}$)

\subsubsection{Substatement}

Det g\"{a}ller att $P\left( \left\vert X\right\vert \leq x\right) =%
%TCIMACRO{\FORMULA{1-\alpha }{1-\alpha }{evaluate}}%
%BeginExpansion
1-\alpha %
%EndExpansion
$. Best\"{a}m $x=$

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{4}$)

\subsubsection{Solution}

Det g\"{a}ller att $X\in N\left( \mu ,\sigma \right) $ varf\"{o}r

\begin{enumerate}
\item $P\left( X\leq 
%TCIMACRO{\FORMULA{a}{a}{evaluate}}%
%BeginExpansion
a%
%EndExpansion
\right) =\frac{1}{\sqrt{%
%TCIMACRO{\FORMULA{2\sigma ^{2}}{2\sigma ^{2}}{evaluate}}%
%BeginExpansion
2\sigma ^{2}%
%EndExpansion
\pi }}\int e^{-\frac{\left( x-%
%TCIMACRO{\FORMULA{\mu }{\mu }{evaluate}}%
%BeginExpansion
\mu %
%EndExpansion
\right) ^{2}}{%
%TCIMACRO{\FORMULA{2\sigma ^{2}}{2\sigma ^{2}}{evaluate}}%
%BeginExpansion
2\sigma ^{2}%
%EndExpansion
}}\,dx=%
%TCIMACRO{%
%\FORMULA{\limfunc{svar}_{1}}{\left( \limfunc{svar}\right) 1}{evaluate}}%
%BeginExpansion
\left( \limfunc{svar}\right) 1%
%EndExpansion
$

\item $P\left( 
%TCIMACRO{\FORMULA{b}{b}{evaluate}}%
%BeginExpansion
b%
%EndExpansion
<X<%
%TCIMACRO{\FORMULA{c}{c}{evaluate}}%
%BeginExpansion
c%
%EndExpansion
\right) =%
%TCIMACRO{%
%\FORMULA{\limfunc{svar}_{2}}{\left( \limfunc{svar}\right) 2}{evaluate}}%
%BeginExpansion
\left( \limfunc{svar}\right) 2%
%EndExpansion
$

\item F\"{o}r att finna $x$ har vi att ta inversa funktionen p\aa\ ekvationen%
\begin{equation*}
\Phi \left( \frac{x-%
%TCIMACRO{\FORMULA{\mu }{\mu }{evaluate}}%
%BeginExpansion
\mu %
%EndExpansion
}{%
%TCIMACRO{\FORMULA{\sigma }{\sigma }{evaluate}}%
%BeginExpansion
\sigma %
%EndExpansion
}\right) =%
%TCIMACRO{\FORMULA{1-\alpha }{1-\alpha }{evaluate}}%
%BeginExpansion
1-\alpha %
%EndExpansion
\end{equation*}%
vilken ger att $x=%
%TCIMACRO{%
%\FORMULA{\limfunc{svar}_{3}}{\left( \limfunc{svar}\right) 3}{evaluate}}%
%BeginExpansion
\left( \limfunc{svar}\right) 3%
%EndExpansion
$.

\item Vi har att%
\begin{equation*}
\Phi \left( \frac{x-%
%TCIMACRO{\FORMULA{\mu }{\mu }{evaluate}}%
%BeginExpansion
\mu %
%EndExpansion
}{%
%TCIMACRO{\FORMULA{\sigma }{\sigma }{evaluate}}%
%BeginExpansion
\sigma %
%EndExpansion
}\right) -\Phi \left( -\frac{x-%
%TCIMACRO{\FORMULA{\mu }{\mu }{evaluate}}%
%BeginExpansion
\mu %
%EndExpansion
}{%
%TCIMACRO{\FORMULA{\sigma }{\sigma }{evaluate}}%
%BeginExpansion
\sigma %
%EndExpansion
}\right) =%
%TCIMACRO{\FORMULA{1-\alpha }{1-\alpha }{evaluate}}%
%BeginExpansion
1-\alpha %
%EndExpansion
\end{equation*}%
varav vi erh\aa ller ekvationen%
\begin{equation*}
\Phi \left( \frac{x-%
%TCIMACRO{\FORMULA{\mu }{\mu }{evaluate}}%
%BeginExpansion
\mu %
%EndExpansion
}{%
%TCIMACRO{\FORMULA{\sigma }{\sigma }{evaluate}}%
%BeginExpansion
\sigma %
%EndExpansion
}\right) =\frac{%
%TCIMACRO{\FORMULA{1-\alpha }{1-\alpha }{evaluate}}%
%BeginExpansion
1-\alpha %
%EndExpansion
+1}{2}=%
%TCIMACRO{\FORMULA{\frac{2-\alpha }{2}}{1-\frac{1}{2}\alpha }{evaluate}}%
%BeginExpansion
1-\frac{1}{2}\alpha %
%EndExpansion
\end{equation*}%
vars l\"{o}sning \"{a}r%
\begin{equation*}
x=%
%TCIMACRO{%
%\FORMULA{\limfunc{svar}_{4}}{\left( \limfunc{svar}\right) 4}{evaluate}}%
%BeginExpansion
\left( \limfunc{svar}\right) 4%
%EndExpansion
\text{.}
\end{equation*}
\end{enumerate}

\subsection{Variant}

\subsubsection{Setup}

$\limfunc{svar}:=0.733$

\subsubsection{Statement}

P\aa\ en rak linje med tre p\aa\ varandra f\"{o}ljande punkter A, B och C m%
\"{a}ts str\"{a}ckorna AB och BC varvid m\"{a}tfelen blir oberoende
stokastiska variabler $X$ s\aa dana att $X_{\text{AB}}\in N\left(
0,0.2\right) $ och $X_{\text{BC}}\in N\left( 0.03\right) $ (sort mm). F\"{o}%
r att ber\"{a}kna str\"{a}ckan AC l\"{a}ggs str\"{a}ckorna AB och BC ihop
och det totala m\"{a}tfelet blir d\"{a}rf\"{o}r summan $X_{AB}+X_{BC}$. Ber%
\"{a}kna sannolikheten att det totala m\"{a}tfelet ligger mellan $-0.4$ och $%
0.4$ (svara med 3 decimaler).

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}$)

\subsubsection{Solution}

S\"{a}tt%
\begin{equation*}
Y=X_{AB}+X_{BC}
\end{equation*}%
d\aa\ g\"{a}ller att $Y\in N\left( 0+0,\sqrt{0.2^{2}+0.3^{2}}\right)
=N\left( 0,\sqrt{0.13}\right) $. S\"{o}kt sannolikhet blir%
\begin{align*}
P\left( -0.4<Z<0.4\right) & =\func{NormalDist}\left( 0.4;0,\sqrt{0.13}\right)
\\
& -\func{NormalDist}\left( -0.4;0,\sqrt{0.13}\right) \\
& =%
%TCIMACRO{\FORMULA{\limfunc{svar}}{\limfunc{svar}}{evaluate}}%
%BeginExpansion
\limfunc{svar}%
%EndExpansion
\end{align*}

\subsection{Variant}

\subsubsection{Setup}

$a:=\limfunc{nplaces}(\func{rand}(99,104)/100,2)$

$b:=\limfunc{nplaces}(\func{rand}(100a-4,100a-1)/100,2)$

$\alpha :=\limfunc{nplaces}(\func{rand}(6,9)/100,2)$

$\beta :=\limfunc{nplaces}(\func{rand}(1,3)/100,2)$

$x:=\func{NormalInv}\left( 1-\alpha ;0,1\right) $

$y:=\func{NormalInv}\left( \beta ;0,1\right) $

$\mu :=\limfunc{nplaces}\left( \frac{bx-ay}{x-y},3\right) $

$\sigma :=\limfunc{nplaces}\left( \frac{a-b}{x-y},3\right) $

$\limfunc{svar}:=\left( \mu ,\sigma \right) $

\subsubsection{Statement}

Vid en kvalitetskontroll av st\aa laxlar sorterar man bort de axlar vars
diameter \"{a}r st\"{o}rre \"{a}n $%
%TCIMACRO{\FORMULA{a}{a}{evaluate}}%
%BeginExpansion
a%
%EndExpansion
$ mm och mindre \"{a}n $%
%TCIMACRO{\FORMULA{b}{b}{evaluate}}%
%BeginExpansion
b%
%EndExpansion
$ mm. S\"{a}tt%
\begin{equation*}
X=\text{axeldiameter i mm.}
\end{equation*}%
Man har genom m\"{a}tningar funnit att%
\begin{equation*}
P\left( X>%
%TCIMACRO{\FORMULA{a}{a}{evaluate}}%
%BeginExpansion
a%
%EndExpansion
\right) =%
%TCIMACRO{\FORMULA{\alpha }{\alpha }{evaluate}}%
%BeginExpansion
\alpha %
%EndExpansion
\text{ samt }P\left( X<%
%TCIMACRO{\FORMULA{b}{b}{evaluate}}%
%BeginExpansion
b%
%EndExpansion
\right) =%
%TCIMACRO{\FORMULA{\beta }{\beta }{evaluate}}%
%BeginExpansion
\beta %
%EndExpansion
\end{equation*}%
Antag att $X\in N\left( \mu ,\sigma \right) $. (svara med tre decimaler)

\subsubsection{Substatement}

Best\"{a}m $\mu =$

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{1}$)

\subsubsection{Substatement}

Best\"{a}m $\sigma =$

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{2}$)

\subsubsection{Solution}

Ekvationssystemet%
\begin{align*}
P\left( X>%
%TCIMACRO{\FORMULA{a}{a}{evaluate}}%
%BeginExpansion
a%
%EndExpansion
\right) & =%
%TCIMACRO{\FORMULA{\alpha }{\alpha }{evaluate} }%
%BeginExpansion
\alpha 
%EndExpansion
\\
P\left( X\leq 
%TCIMACRO{\FORMULA{b}{b}{evaluate}}%
%BeginExpansion
b%
%EndExpansion
\right) & =%
%TCIMACRO{\FORMULA{\beta }{\beta }{evaluate}}%
%BeginExpansion
\beta %
%EndExpansion
\end{align*}%
f\"{o}ljer ur uppgifterna och det kan skrivas%
\begin{align*}
P\left( \frac{X-\mu }{\sigma }\leq \frac{%
%TCIMACRO{\FORMULA{a}{a}{evaluate}}%
%BeginExpansion
a%
%EndExpansion
-\mu }{\sigma }\right) & =%
%TCIMACRO{\FORMULA{1-\alpha }{1-\alpha }{evaluate} }%
%BeginExpansion
1-\alpha 
%EndExpansion
\\
P\left( \frac{X-\mu }{\sigma }\leq \frac{%
%TCIMACRO{\FORMULA{b}{b}{evaluate}}%
%BeginExpansion
b%
%EndExpansion
-\mu }{\sigma }\right) & =%
%TCIMACRO{\FORMULA{\beta }{\beta }{evaluate}}%
%BeginExpansion
\beta %
%EndExpansion
\end{align*}%
G\aa\ nu \"{o}ver till Scientific NoteBook:s beteckningar%
\begin{align*}
\Phi \left( \frac{%
%TCIMACRO{\FORMULA{a}{a}{evaluate}}%
%BeginExpansion
a%
%EndExpansion
-\mu }{\sigma }\right) & =%
%TCIMACRO{\FORMULA{1-\alpha }{1-\alpha }{evaluate} }%
%BeginExpansion
1-\alpha 
%EndExpansion
\\
\Phi \left( \frac{%
%TCIMACRO{\FORMULA{b}{b}{evaluate}}%
%BeginExpansion
b%
%EndExpansion
-\mu }{\sigma }\right) & =%
%TCIMACRO{\FORMULA{\beta }{\beta }{evaluate}}%
%BeginExpansion
\beta %
%EndExpansion
\end{align*}%
och ta inversa funktionen p\aa\ b\aa da sidor%
\begin{align*}
%TCIMACRO{\U{b4}}%
%BeginExpansion
{\acute{}}%
%EndExpansion
\Phi ^{-1}\left( \Phi \left( \frac{%
%TCIMACRO{\FORMULA{a}{a}{evaluate}}%
%BeginExpansion
a%
%EndExpansion
-\mu }{\sigma }\right) \right) & =\Phi ^{-1}\left( 
%TCIMACRO{\FORMULA{1-\alpha }{1-\alpha }{evaluate}}%
%BeginExpansion
1-\alpha %
%EndExpansion
\right) \\
\Phi ^{-1}\left( \Phi \left( \frac{%
%TCIMACRO{\FORMULA{b}{b}{evaluate}}%
%BeginExpansion
b%
%EndExpansion
-\mu }{\sigma }\right) \right) & =\Phi ^{-1}\left( 
%TCIMACRO{\FORMULA{\beta }{\beta }{evaluate}}%
%BeginExpansion
\beta %
%EndExpansion
\right)
\end{align*}%
f\"{o}r att slutligen erh\aa lla ekvationssystemet%
\begin{align*}
\frac{%
%TCIMACRO{\FORMULA{a}{a}{evaluate}}%
%BeginExpansion
a%
%EndExpansion
-\mu }{\sigma }& =%
%TCIMACRO{\FORMULA{x}{x}{evaluate} }%
%BeginExpansion
x
%EndExpansion
\\
\frac{%
%TCIMACRO{\FORMULA{b}{b}{evaluate}}%
%BeginExpansion
b%
%EndExpansion
-\mu }{\sigma }& =%
%TCIMACRO{\FORMULA{y}{y}{evaluate}}%
%BeginExpansion
y%
%EndExpansion
\end{align*}%
vilket ger $\mu =\frac{%
%TCIMACRO{\FORMULA{b}{b}{evaluate}}%
%BeginExpansion
b%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{x}{x}{evaluate}}%
%BeginExpansion
x%
%EndExpansion
-%
%TCIMACRO{\FORMULA{a}{a}{evaluate}}%
%BeginExpansion
a%
%EndExpansion
\times \left( 
%TCIMACRO{\FORMULA{y}{y}{evaluate}}%
%BeginExpansion
y%
%EndExpansion
\right) }{%
%TCIMACRO{\FORMULA{x}{x}{evaluate}}%
%BeginExpansion
x%
%EndExpansion
-\left( 
%TCIMACRO{\FORMULA{y}{y}{evaluate}}%
%BeginExpansion
y%
%EndExpansion
\right) }=%
%TCIMACRO{%
%\FORMULA{\limfunc{svar}_{1}}{\left( \limfunc{svar}\right) 1}{evaluate}}%
%BeginExpansion
\left( \limfunc{svar}\right) 1%
%EndExpansion
$ och $\sigma =\frac{%
%TCIMACRO{\FORMULA{a}{a}{evaluate}}%
%BeginExpansion
a%
%EndExpansion
-%
%TCIMACRO{\FORMULA{b}{b}{evaluate}}%
%BeginExpansion
b%
%EndExpansion
}{%
%TCIMACRO{\FORMULA{x}{x}{evaluate}}%
%BeginExpansion
x%
%EndExpansion
-\left( 
%TCIMACRO{\FORMULA{y}{y}{evaluate}}%
%BeginExpansion
y%
%EndExpansion
\right) }=%
%TCIMACRO{%
%\FORMULA{\limfunc{svar}_{2}}{\left( \limfunc{svar}\right) 2}{evaluate}}%
%BeginExpansion
\left( \limfunc{svar}\right) 2%
%EndExpansion
$.

\subsection{Variant}

\subsubsection{Setup}

$\mu :=\left( \func{rand}\left( 1,5\right) ,\func{rand}\left( 3,6\right)
\right) $

$\sigma :=(\func{rand}(1,2),\func{rand}(4,6))$

$n:=(\func{rand}(1,4),\func{rand}(2,6))$

$z:=\limfunc{nplaces}\left( \func{NormalInv}\left( \func{rand}%
(1000000)/1000000.0;n_{1}\mu _{1}+n_{2}\mu _{2},\sqrt{n_{1}^{2}\sigma
_{1}^{2}+n_{2}^{2}\sigma _{2}^{2}}\right) ,2\right) $

$\lambda :=\frac{z-n_{1}\mu _{1}-n_{2}\mu _{2}}{\sqrt{n_{1}^{2}\sigma
_{1}^{2}+n_{2}^{2}\sigma _{2}^{2}}}$

$\limfunc{svar}:=\limfunc{nplaces}\left( \func{NormalDist}\left( \lambda
;0,1\right) ,2\right) $

\subsubsection{Statement}

De oberoende stokastiska variablerna $X$ och $Y$ \"{a}r normalf\"{o}rdelade
-- $X\in N\left( 
%TCIMACRO{\FORMULA{\mu _{1}}{\mu _{1}}{evaluate}}%
%BeginExpansion
\mu _{1}%
%EndExpansion
,%
%TCIMACRO{\FORMULA{\sigma _{1}}{\sigma _{1}}{evaluate}}%
%BeginExpansion
\sigma _{1}%
%EndExpansion
\right) $ och $Y\in N\left( 
%TCIMACRO{\FORMULA{\mu _{2}}{\mu _{2}}{evaluate}}%
%BeginExpansion
\mu _{2}%
%EndExpansion
,%
%TCIMACRO{\FORMULA{\sigma _{2}}{\sigma _{2}}{evaluate}}%
%BeginExpansion
\sigma _{2}%
%EndExpansion
\right) $. Ber\"{a}kna, med tv\aa\ decimalers noggranhet, 
\begin{equation*}
P\left( 
%TCIMACRO{\FORMULA{n_{1}}{n_{1}}{evaluate}}%
%BeginExpansion
n_{1}%
%EndExpansion
X+%
%TCIMACRO{\FORMULA{n_{2}}{n_{2}}{evaluate}}%
%BeginExpansion
n_{2}%
%EndExpansion
Y\leq 
%TCIMACRO{\FORMULA{z}{z}{evaluate}}%
%BeginExpansion
z%
%EndExpansion
\right)
\end{equation*}

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}$)

\paragraph{Solution}

En summa av normalf\"{o}rdelade stokastiska variabler \"{a}r normalf\"{o}%
rdelad och s\aa ledes har vi att ber\"{a}kna v\"{a}ntev\"{a}rde och
standardavvikelse f\"{o}r den nya variabeln. 
\begin{align*}
E\left( 
%TCIMACRO{\FORMULA{n_{1}}{n_{1}}{evaluate}}%
%BeginExpansion
n_{1}%
%EndExpansion
X+%
%TCIMACRO{\FORMULA{n_{2}}{n_{2}}{evaluate}}%
%BeginExpansion
n_{2}%
%EndExpansion
Y\right) & =%
%TCIMACRO{\FORMULA{n_{1}}{n_{1}}{evaluate}}%
%BeginExpansion
n_{1}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{\mu _{1}}{\mu _{1}}{evaluate}}%
%BeginExpansion
\mu _{1}%
%EndExpansion
+%
%TCIMACRO{\FORMULA{n_{2}}{n_{2}}{evaluate}}%
%BeginExpansion
n_{2}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{\mu _{2}}{\mu _{2}}{evaluate}}%
%BeginExpansion
\mu _{2}%
%EndExpansion
=%
%TCIMACRO{%
%\FORMULA{n_{1}\mu _{1}+n_{2}\mu _{2}}{\mu _{1}n_{1}+\mu _{2}n_{2}}{evaluate} }%
%BeginExpansion
\mu _{1}n_{1}+\mu _{2}n_{2}
%EndExpansion
\\
V\left( 
%TCIMACRO{\FORMULA{n_{1}}{n_{1}}{evaluate}}%
%BeginExpansion
n_{1}%
%EndExpansion
X+%
%TCIMACRO{\FORMULA{n_{2}}{n_{2}}{evaluate}}%
%BeginExpansion
n_{2}%
%EndExpansion
Y\right) & =%
%TCIMACRO{\FORMULA{n_{1}^{2}}{n_{1}^{2}}{evaluate}}%
%BeginExpansion
n_{1}^{2}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{\sigma _{1}^{2}}{\sigma _{1}^{2}}{evaluate}}%
%BeginExpansion
\sigma _{1}^{2}%
%EndExpansion
+%
%TCIMACRO{\FORMULA{n_{2}^{2}}{n_{2}^{2}}{evaluate}}%
%BeginExpansion
n_{2}^{2}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{\sigma _{2}^{2}}{\sigma _{2}^{2}}{evaluate}}%
%BeginExpansion
\sigma _{2}^{2}%
%EndExpansion
=%
%TCIMACRO{%
%\FORMULA{n_{1}^{2}\sigma _{1}^{2}+n_{2}^{2}\sigma _{2}^{2}}{\sigma _{1}^{2}n_{1}^{2}+\sigma _{2}^{2}n_{2}^{2}}{evaluate} }%
%BeginExpansion
\sigma _{1}^{2}n_{1}^{2}+\sigma _{2}^{2}n_{2}^{2}
%EndExpansion
\\
D\left( 
%TCIMACRO{\FORMULA{n_{1}}{n_{1}}{evaluate}}%
%BeginExpansion
n_{1}%
%EndExpansion
X+%
%TCIMACRO{\FORMULA{n_{2}}{n_{2}}{evaluate}}%
%BeginExpansion
n_{2}%
%EndExpansion
Y\right) & =\sqrt{%
%TCIMACRO{%
%\FORMULA{n_{1}^{2}\sigma _{1}^{2}+n_{2}^{2}\sigma _{2}^{2}}{\sigma _{1}^{2}n_{1}^{2}+\sigma _{2}^{2}n_{2}^{2}}{evaluate}}%
%BeginExpansion
\sigma _{1}^{2}n_{1}^{2}+\sigma _{2}^{2}n_{2}^{2}%
%EndExpansion
}
\end{align*}%
Varav vi erh\aa ller 
\begin{align*}
P\left( 
%TCIMACRO{\FORMULA{n_{1}}{n_{1}}{evaluate}}%
%BeginExpansion
n_{1}%
%EndExpansion
X+%
%TCIMACRO{\FORMULA{n_{2}}{n_{2}}{evaluate}}%
%BeginExpansion
n_{2}%
%EndExpansion
Y\leq 
%TCIMACRO{\FORMULA{z}{z}{evaluate}}%
%BeginExpansion
z%
%EndExpansion
\right) & =P\left( \frac{%
%TCIMACRO{\FORMULA{n_{1}}{n_{1}}{evaluate}}%
%BeginExpansion
n_{1}%
%EndExpansion
X+%
%TCIMACRO{\FORMULA{n_{2}}{n_{2}}{evaluate}}%
%BeginExpansion
n_{2}%
%EndExpansion
Y-%
%TCIMACRO{%
%\FORMULA{n_{1}\mu _{1}+n_{2}\mu _{2}}{\mu _{1}n_{1}+\mu _{2}n_{2}}{evaluate}}%
%BeginExpansion
\mu _{1}n_{1}+\mu _{2}n_{2}%
%EndExpansion
}{\sqrt{%
%TCIMACRO{%
%\FORMULA{n_{1}^{2}\sigma _{1}^{2}+n_{2}^{2}\sigma _{2}^{2}}{\sigma _{1}^{2}n_{1}^{2}+\sigma _{2}^{2}n_{2}^{2}}{evaluate}}%
%BeginExpansion
\sigma _{1}^{2}n_{1}^{2}+\sigma _{2}^{2}n_{2}^{2}%
%EndExpansion
}}\leq \frac{%
%TCIMACRO{\FORMULA{z}{z}{evaluate}}%
%BeginExpansion
z%
%EndExpansion
-%
%TCIMACRO{%
%\FORMULA{n_{1}\mu _{1}+n_{2}\mu _{2}}{\mu _{1}n_{1}+\mu _{2}n_{2}}{evaluate}}%
%BeginExpansion
\mu _{1}n_{1}+\mu _{2}n_{2}%
%EndExpansion
}{\sqrt{%
%TCIMACRO{%
%\FORMULA{n_{1}^{2}\sigma _{1}^{2}+n_{2}^{2}\sigma _{2}^{2}}{\sigma _{1}^{2}n_{1}^{2}+\sigma _{2}^{2}n_{2}^{2}}{evaluate}}%
%BeginExpansion
\sigma _{1}^{2}n_{1}^{2}+\sigma _{2}^{2}n_{2}^{2}%
%EndExpansion
}}\right) \\
& =\Phi \left( 
%TCIMACRO{\FORMULA{\lambda }{\lambda }{evaluate}}%
%BeginExpansion
\lambda %
%EndExpansion
\right) \\
& \approx 
%TCIMACRO{\FORMULA{\limfunc{svar}}{\limfunc{svar}}{evaluate}}%
%BeginExpansion
\limfunc{svar}%
%EndExpansion
\end{align*}

\subsection{Variant}

\subsubsection{Setup}

$\limfunc{svar}:=1-e^{-2\left( \alpha x\right) ^{\beta }}$

\subsubsection{Statement}

Ett seriekopplat system best\aa r av tv\aa\ komponenter som fungerar
oberoende av varandra och b\aa da \"{a}r $Wei\left( \alpha ,\beta \right) $.
Best\"{a}m t\"{a}thetsfunktionen f\"{o}r systemets livsl\"{a}ngd.

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}$)

\subsubsection{Solution}

S\"{a}tt%
\begin{align*}
X_{1}& =\text{komponent }1\text{:s livsl\"{a}ngd} \\
X_{2}& =\text{komponent }2\text{:s livsl\"{a}ngd} \\
Z& =\min \left( X_{1},X_{2}\right) =\text{systemets livsl\"{a}ngd}
\end{align*}%
det g\"{a}ller att $X_{i}\in Wei\left( \alpha ,\beta \right) $. F\"{o}%
rdelningsfunktionen f\"{o}r $Z$ blir%
\begin{align*}
P\left( Z\leq z\right) & =P\left( \min \left( X_{1},X_{2}\right) \leq
z\right) \\
& =1-P\left( \min \left( X_{1},X_{2}\right) >z\right) \\
\left\{ \text{ty oberoende}\right\} & =1-P\left( X_{1}>z\right) P\left(
X_{2}>z\right) \\
& =1-e^{-\left( \alpha x\right) ^{\beta }}e^{-\left( \alpha x\right) ^{\beta
}} \\
& =1-e^{-2\left( \alpha x\right) ^{\beta }}
\end{align*}%
varf\"{o}r $Z\in Wei\left( 2^{1/\beta }\alpha ,\beta \right) $.

\section{Question}

\subsection{Comment}

Approximationer

\subsection{Variant}

\subsubsection{Setup}

$\lambda :=\left( \func{rand}\left( 36,44\right) ,\func{rand}\left(
8,12\right) ,\func{rand}\left( 10,14\right) \right) $

$z:=-\frac{2\left( \lambda _{2}+\lambda _{3}\right) -\lambda _{1}}{\sqrt{%
4.0\left( \lambda _{2}+\lambda _{3}\right) +\lambda _{1}}}$

$p:=\func{NormalDist}\left( z;0,1\right) $

$\limfunc{svar}:=\limfunc{nplaces}\left( p,3\right) $

\subsubsection{Statement}

Beteckna antalet uppdrag under en m\aa nad f\"{o}r tre kunder med $%
X_{1},X_{2}$ och $X_{3}$. Dessa uppdrag betraktas som oberoende stokastiska
variabler s\aa dana att $X_{1}\in Po\left( 
%TCIMACRO{\FORMULA{\lambda _{1}}{\lambda _{1}}{evaluate}}%
%BeginExpansion
\lambda _{1}%
%EndExpansion
\right) ,X_{2}\in Po\left( 
%TCIMACRO{\FORMULA{\lambda _{2}}{\lambda _{2}}{evaluate}}%
%BeginExpansion
\lambda _{2}%
%EndExpansion
\right) $ och $X_{3}\in Po\left( 
%TCIMACRO{\FORMULA{\lambda _{3}}{\lambda _{3}}{evaluate}}%
%BeginExpansion
\lambda _{3}%
%EndExpansion
\right) $. Ber\"{a}kna sannolikheten att den f\"{o}rsta kunden ger mer \"{a}%
n dubbelt s\aa\ m\aa nga uppdrag som de andra tv\aa\ tillsammans (svara med
tre decimaler).

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}$)

\subsubsection{Solution}

De tv\aa\ sista kunderna ger uppdrag svarande mot en Poissonf\"{o}rdelning
med parametern $%
%TCIMACRO{%
%\FORMULA{\lambda _{2}+\lambda _{3}}{\lambda _{2}+\lambda _{3}}{evaluate}}%
%BeginExpansion
\lambda _{2}+\lambda _{3}%
%EndExpansion
$. Tumregeln f\"{o}r normalapproximation av Poissonf\"{o}rdelningen \"{a}r v%
\"{a}l uppfylld och d\"{a}rf\"{o}r g\"{a}ller%
\begin{align*}
X_{1}& \approx N\left( 
%TCIMACRO{\FORMULA{\lambda _{1}}{\lambda _{1}}{evaluate}}%
%BeginExpansion
\lambda _{1}%
%EndExpansion
,\sqrt{%
%TCIMACRO{\FORMULA{\lambda _{1}}{\lambda _{1}}{evaluate}}%
%BeginExpansion
\lambda _{1}%
%EndExpansion
}\right) \\
X_{2}+X_{3}& \approx N\left( 
%TCIMACRO{%
%\FORMULA{\lambda _{2}+\lambda _{3}}{\lambda _{2}+\lambda _{3}}{evaluate}}%
%BeginExpansion
\lambda _{2}+\lambda _{3}%
%EndExpansion
,\sqrt{%
%TCIMACRO{%
%\FORMULA{\lambda _{2}+\lambda _{3}}{\lambda _{2}+\lambda _{3}}{evaluate}}%
%BeginExpansion
\lambda _{2}+\lambda _{3}%
%EndExpansion
}\right)
\end{align*}%
Den s\"{o}kta sannolikheten kan nu skrivas%
\begin{align*}
P\left( X_{1}-2\left( X_{2}+X_{3}\right) >0\right) & =P\left( 2\left(
X_{2}+X_{3}\right) -X_{1}<0\right) \\
& \approx \Phi \left( -\frac{2\times 
%TCIMACRO{%
%\FORMULA{\lambda _{2}+\lambda _{3}}{\lambda _{2}+\lambda _{3}}{evaluate}}%
%BeginExpansion
\lambda _{2}+\lambda _{3}%
%EndExpansion
-%
%TCIMACRO{\FORMULA{\lambda _{1}}{\lambda _{1}}{evaluate}}%
%BeginExpansion
\lambda _{1}%
%EndExpansion
}{\sqrt{2^{2}\times 
%TCIMACRO{%
%\FORMULA{\lambda _{2}+\lambda _{3}}{\lambda _{2}+\lambda _{3}}{evaluate}}%
%BeginExpansion
\lambda _{2}+\lambda _{3}%
%EndExpansion
+%
%TCIMACRO{\FORMULA{\lambda _{1}}{\lambda _{1}}{evaluate}}%
%BeginExpansion
\lambda _{1}%
%EndExpansion
}}\right) \\
& \approx 
%TCIMACRO{\FORMULA{\limfunc{svar}}{\limfunc{svar}}{evaluate}}%
%BeginExpansion
\limfunc{svar}%
%EndExpansion
\end{align*}

\subsection{Variant}

\subsubsection{Setup}

$\lambda :=\func{rand}(300,500)$

$p:=\limfunc{nplaces}(\func{rand}(100,500)/10000,3)$

$\limfunc{svar}:=\left\lfloor \lambda +\func{NormalInv}\left( 1-p;0,1\right)
\times \sqrt{\lambda }\right\rfloor $

\subsubsection{Statement}

Antalet passagerare som \"{o}nskar \aa ka med ett visst t\aa g kan betraktas
som en stokastisk variabel som \"{a}r Poissonf\"{o}rdelad med parametern $%
\lambda =%
%TCIMACRO{\FORMULA{\lambda }{\lambda }{evaluate}}%
%BeginExpansion
\lambda %
%EndExpansion
$. Hur m\aa nga platser beh\"{o}ver man ha i t\aa get f\"{o}r att
sannolikheten att det skall bli fullsatt skall vara h\"{o}gst $%
%TCIMACRO{\FORMULA{p}{p}{evaluate}}%
%BeginExpansion
p%
%EndExpansion
$ (svara med $0$ decimaler).

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}$)

\paragraph{Solution}

S\"{a}tt 
\begin{equation*}
X=\text{antalet passagerare}\in Po\left( 
%TCIMACRO{\FORMULA{\lambda }{\lambda }{evaluate}}%
%BeginExpansion
\lambda %
%EndExpansion
\right)
\end{equation*}%
d\aa\ erh\aa lls med $n=$antal platser att 
\begin{align*}
P\left( X\geq n\right) & <%
%TCIMACRO{\FORMULA{p}{p}{evaluate} }%
%BeginExpansion
p
%EndExpansion
\\
\sum_{k=n}^{\infty }\frac{400^{k}}{k!}e^{-400}& <%
%TCIMACRO{\FORMULA{p}{p}{evaluate} }%
%BeginExpansion
p
%EndExpansion
\\
\sum_{k=0}^{n-1}\frac{400^{k}}{k!}e^{-400}& >%
%TCIMACRO{\FORMULA{1-p}{1-p}{evaluate}}%
%BeginExpansion
1-p%
%EndExpansion
\end{align*}

Scientific NoteBook g\"{o}r det m\"{o}jligt att pr\"{o}va sig fram \"{a}ven
om det tar lite tid. En annan l\"{o}sning \"{a}r att notera att $400>15$ och
normalapproximation kan d\"{a}rf\"{o}r anv\"{a}ndas: 
\begin{align*}
P\left( X\geq n\right) & \approx 1-\func{NormalDist}\left( \frac{n-%
%TCIMACRO{\FORMULA{\lambda }{\lambda }{evaluate}}%
%BeginExpansion
\lambda %
%EndExpansion
}{\sqrt{%
%TCIMACRO{\FORMULA{\lambda }{\lambda }{evaluate}}%
%BeginExpansion
\lambda %
%EndExpansion
}};0,1\right) \\
\func{NormalDist}\left( \frac{n-%
%TCIMACRO{\FORMULA{\lambda }{\lambda }{evaluate}}%
%BeginExpansion
\lambda %
%EndExpansion
}{\sqrt{%
%TCIMACRO{\FORMULA{\lambda }{\lambda }{evaluate}}%
%BeginExpansion
\lambda %
%EndExpansion
}};0,1\right) & \approx 
%TCIMACRO{\FORMULA{1-p}{1-p}{evaluate}}%
%BeginExpansion
1-p%
%EndExpansion
\end{align*}

vilket ger ekvationen 
\begin{equation*}
\frac{n-%
%TCIMACRO{\FORMULA{\lambda }{\lambda }{evaluate}}%
%BeginExpansion
\lambda %
%EndExpansion
}{\sqrt{%
%TCIMACRO{\FORMULA{\lambda }{\lambda }{evaluate}}%
%BeginExpansion
\lambda %
%EndExpansion
}}\approx \func{NormalInv}\left( 
%TCIMACRO{\FORMULA{1-p}{1-p}{evaluate}}%
%BeginExpansion
1-p%
%EndExpansion
;0,1\right) =%
%TCIMACRO{%
%\FORMULA{\func{NormalInv}\left( 1-p;0,1\right) }{\left( \func{stats}\func{normalQuantile}\left( 0,1\right) \right) \left( 1-p\right) }{evaluate}}%
%BeginExpansion
\left( \func{stats}\func{normalQuantile}\left( 0,1\right) \right) \left( 1-p\right) %
%EndExpansion
\end{equation*}

vars l\"{o}sning \"{a}r $n\approx \left\lfloor 
%TCIMACRO{\FORMULA{\lambda }{\lambda }{evaluate}}%
%BeginExpansion
\lambda %
%EndExpansion
+%
%TCIMACRO{%
%\FORMULA{\func{NormalInv}\left( 1-p;0,1\right) }{\left( \func{stats}\func{normalQuantile}\left( 0,1\right) \right) \left( 1-p\right) }{evaluate}}%
%BeginExpansion
\left( \func{stats}\func{normalQuantile}\left( 0,1\right) \right) \left( 1-p\right) %
%EndExpansion
\times \sqrt{%
%TCIMACRO{\FORMULA{\lambda }{\lambda }{evaluate}}%
%BeginExpansion
\lambda %
%EndExpansion
}\right\rfloor =%
%TCIMACRO{\FORMULA{\limfunc{svar}}{\limfunc{svar}}{evaluate}}%
%BeginExpansion
\limfunc{svar}%
%EndExpansion
$

\subsection{Variant}

\subsubsection{Setup}

$\mu :=\func{rand}(450,550)/1000$

$n:=\func{rand}\left( 75,125\right) $

$z:=\func{rand}\left( 1000\mu -50,1000\mu -50\right) /10$

$\limfunc{svar}:=\limfunc{nplaces}\left( 1-\func{NormalDist}\left( \frac{%
1.0z-n\mu }{\sqrt{n\mu ^{2}}};0,1\right) ,3\right) $

\subsubsection{Statement}

Vid testning av en dator placeras den i en milj\"{o} d\"{a}r den uts\"{a}tts
f\"{o}r st\"{o}rningar. Tidsavst\aa ndet mellan tv\aa\ p\aa\ varandra f\"{o}%
ljande st\"{o}rningar \"{a}r exponentialf\"{o}rdelad med v\"{a}ntev\"{a}rdet 
$%
%TCIMACRO{\FORMULA{\mu }{\mu }{evaluatenum}}%
%BeginExpansion
\mu %
%EndExpansion
$ minuter. Man r\"{a}knar med att maskinen g\aa r s\"{o}nder vid den $%
%TCIMACRO{\FORMULA{n}{n}{evaluate}}%
%BeginExpansion
n%
%EndExpansion
$:e st\"{o}rningen samt att st\"{o}rningarna sker oberoende av varandra. L%
\aa t $Y$ vara tiden fram till och med den $%
%TCIMACRO{\FORMULA{n}{n}{evaluate}}%
%BeginExpansion
n%
%EndExpansion
$:e st\"{o}rningen. Best\"{a}m, med tre decimalers noggranhet, sannolikheten 
$P\left( Y>%
%TCIMACRO{\FORMULA{z}{z}{evaluate}}%
%BeginExpansion
z%
%EndExpansion
\right) $.

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}$)

\subsubsection{Solution}

Eftersom vi har en summa av $%
%TCIMACRO{\FORMULA{n}{n}{evaluate}}%
%BeginExpansion
n%
%EndExpansion
$ oberoende exponentialf\"{o}rdelade variabler med v\"{a}ntev\"{a}rdet $%
%TCIMACRO{\FORMULA{\mu }{\mu }{evaluatenum}}%
%BeginExpansion
\mu %
%EndExpansion
$ till\"{a}mpar vi normalapproximation enligt CGS och erh\aa ller 
\begin{equation*}
P\left( Y>%
%TCIMACRO{\FORMULA{z}{z}{evaluate}}%
%BeginExpansion
z%
%EndExpansion
\right) \approx 1-\Phi \left( \frac{%
%TCIMACRO{\FORMULA{z}{z}{evaluate}}%
%BeginExpansion
z%
%EndExpansion
-%
%TCIMACRO{\FORMULA{n\mu }{n\mu }{evaluate}}%
%BeginExpansion
n\mu %
%EndExpansion
}{\sqrt{%
%TCIMACRO{\FORMULA{n\mu ^{2}}{n\mu ^{2}}{evaluate}}%
%BeginExpansion
n\mu ^{2}%
%EndExpansion
}}\right) =%
%TCIMACRO{\FORMULA{\limfunc{svar}}{\limfunc{svar}}{evaluate}}%
%BeginExpansion
\limfunc{svar}%
%EndExpansion
\end{equation*}

\subsection{Variant}

\subsubsection{Setup}

$n:=\func{rand}\left( 1400,1600\right) $

$m:=\func{rand}\left( 10,20\right) $

$p:=\func{rand}\left( \left\{ 0.9,0.95,0.99\right\} \right) $

$x:=\func{NormalInv}\left( \frac{1+p}{2};0,1\right) $

$\limfunc{svar}:=\left( \limfunc{nplaces}\left( 1-\func{NormalDist}\left(
m;0,\sqrt{\frac{n}{12}}\right) +\func{NormalDist}\left( -m;0,\sqrt{\frac{n}{%
12}}\right) ,2\right) ,\left\lceil \frac{12\left( m-5\right) ^{2}}{x^{2}}%
\right\rceil \right) $

\subsubsection{Statement}

I ett program avrundas varje tal till n\"{a}rmaste heltal. F\"{o}r
avrundningsfelen g\"{a}ller att de \"{a}r oberoende och likaf\"{o}rdelade
variabler p\aa\ intervallet $\left( -0.5,0.5\right) $.

\subsubsection{Substatement}

Best\"{a}m sannolikheten att absoluta beloppet av totala felet \"{o}%
verstiger $%
%TCIMACRO{\FORMULA{m}{m}{evaluatenum}}%
%BeginExpansion
m%
%EndExpansion
$ n\"{a}r $%
%TCIMACRO{\FORMULA{n}{n}{evaluatenum}}%
%BeginExpansion
n%
%EndExpansion
$ tal adderas (svara med 2 decimaler).

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{1}$)

\subsubsection{Substatement}

Best\"{a}m hur m\aa nga tal som kan adderas f\"{o}r att absoluta beloppet av
totala felet med sannolikheten $%
%TCIMACRO{\FORMULA{p}{p}{evaluatenum}}%
%BeginExpansion
p%
%EndExpansion
$ skall bli mindre \"{a}n $%
%TCIMACRO{\FORMULA{m-5}{m-5.0}{evaluatenum}}%
%BeginExpansion
m-5.0%
%EndExpansion
$ (svara med $0$ decimaler).

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{2}$)

\subsubsection{Solution}

S\"{a}tt 
\begin{equation*}
X_{i}=\text{avrundningsfel tal }i
\end{equation*}%
d\aa\ g\"{a}ller att 
\begin{equation*}
Y_{n}=\sum_{i=1}^{n}X_{i}\approx N\left( 0,\sqrt{\frac{n}{12}}\right)
\end{equation*}%
varf\"{o}r 
\begin{align*}
P\left( \left\vert Y_{%
%TCIMACRO{\FORMULA{n}{n}{evaluatenum}}%
%BeginExpansion
n%
%EndExpansion
}\right\vert >%
%TCIMACRO{\FORMULA{m}{m}{evaluatenum}}%
%BeginExpansion
m%
%EndExpansion
\right) & =1-P\left( \left\vert Y_{%
%TCIMACRO{\FORMULA{n}{n}{evaluatenum}}%
%BeginExpansion
n%
%EndExpansion
}\right\vert \leq 
%TCIMACRO{\FORMULA{m}{m}{evaluatenum}}%
%BeginExpansion
m%
%EndExpansion
\right) \\
& =1-\func{NormalDist}\left( 
%TCIMACRO{\FORMULA{m}{m}{evaluatenum}}%
%BeginExpansion
m%
%EndExpansion
;0,\sqrt{\frac{%
%TCIMACRO{\FORMULA{n}{n}{evaluatenum}}%
%BeginExpansion
n%
%EndExpansion
}{12}}\right) +\func{NormalDist}\left( -%
%TCIMACRO{\FORMULA{m}{m}{evaluatenum}}%
%BeginExpansion
m%
%EndExpansion
;0,\sqrt{\frac{%
%TCIMACRO{\FORMULA{n}{n}{evaluatenum}}%
%BeginExpansion
n%
%EndExpansion
}{12}}\right) \\
& =%
%TCIMACRO{%
%\FORMULA{\limfunc{svar}_{1}}{\left( \limfunc{svar}\right) 1}{evaluatenum}}%
%BeginExpansion
\left( \limfunc{svar}\right) 1%
%EndExpansion
\end{align*}%
F\"{o}r den andra delen g\"{a}ller att $P\left( \left\vert Y_{n}\right\vert <%
%TCIMACRO{\FORMULA{m-5}{m-5.0}{evaluatenum}}%
%BeginExpansion
m-5.0%
%EndExpansion
\right) \geq 
%TCIMACRO{\FORMULA{p}{p}{evaluatenum}}%
%BeginExpansion
p%
%EndExpansion
$ varf\"{o}r%
\begin{align*}
\func{NormalDist}\left( 
%TCIMACRO{\FORMULA{m-5}{m-5.0}{evaluatenum}}%
%BeginExpansion
m-5.0%
%EndExpansion
;0,\sqrt{\frac{n}{12}}\right) -\func{NormalDist}\left( -%
%TCIMACRO{\FORMULA{m-5}{m-5.0}{evaluatenum}}%
%BeginExpansion
m-5.0%
%EndExpansion
;0,\sqrt{\frac{n}{12}}\right) & \geq 
%TCIMACRO{\FORMULA{p}{p}{evaluatenum} }%
%BeginExpansion
p
%EndExpansion
\\
-\func{NormalDist}\left( -\frac{%
%TCIMACRO{\FORMULA{m-5}{m-5.0}{evaluatenum}}%
%BeginExpansion
m-5.0%
%EndExpansion
}{\sqrt{\frac{n}{12}}};0,1\right) & \geq 
%TCIMACRO{\FORMULA{p}{p}{evaluatenum} }%
%BeginExpansion
p
%EndExpansion
\\
2\func{NormalDist}\left( \frac{%
%TCIMACRO{\FORMULA{m-5}{m-5.0}{evaluatenum}}%
%BeginExpansion
m-5.0%
%EndExpansion
}{\sqrt{\frac{n}{12}}};0,1\right) & \geq 
%TCIMACRO{\FORMULA{1+p}{p+1.0}{evaluatenum}}%
%BeginExpansion
p+1.0%
%EndExpansion
\end{align*}%
varf\"{o}r 
\begin{align*}
\func{NormalDist}\left( \frac{%
%TCIMACRO{\FORMULA{m-5}{m-5.0}{evaluatenum}}%
%BeginExpansion
m-5.0%
%EndExpansion
}{\sqrt{\frac{n}{12}}};0,1\right) & \geq 
%TCIMACRO{\FORMULA{\frac{1+p}{2}}{0.5p+0.5}{evaluatenum} }%
%BeginExpansion
0.5p+0.5
%EndExpansion
\\
\frac{%
%TCIMACRO{\FORMULA{m-5}{m-5.0}{evaluatenum}}%
%BeginExpansion
m-5.0%
%EndExpansion
}{\sqrt{\frac{n}{12}}}& \geq \func{NormalInv}\left( 
%TCIMACRO{\FORMULA{\frac{1+p}{2}}{0.5p+0.5}{evaluatenum}}%
%BeginExpansion
0.5p+0.5%
%EndExpansion
;0,1\right) =%
%TCIMACRO{\FORMULA{x}{x}{evaluatenum} }%
%BeginExpansion
x
%EndExpansion
\\
\sqrt{n}& \leq \frac{\sqrt{12}\times 
%TCIMACRO{\FORMULA{m-5}{m-5.0}{evaluatenum}}%
%BeginExpansion
m-5.0%
%EndExpansion
}{%
%TCIMACRO{\FORMULA{x}{x}{evaluatenum}}%
%BeginExpansion
x%
%EndExpansion
} \\
n& \leq 
%TCIMACRO{%
%\FORMULA{\limfunc{svar}_{2}}{\left( \limfunc{svar}\right) 2}{evaluatenum}}%
%BeginExpansion
\left( \limfunc{svar}\right) 2%
%EndExpansion
\end{align*}

\subsection{Variant}

\subsubsection{Setup}

$\mu :=\left( \func{rand}(12,14)/10,\func{rand}(90,110)/10\right) $

$\sigma :=\left( \func{rand}(4,6)/10,\func{rand}(28,32)/10\right) $

$n:=\left( \func{rand}\left( 34,38\right) ,\func{rand}\left( 23,27\right)
\right) $

$a:=\func{rand}\left( 300,330\right) $

$m:=n_{1}\mu _{1}+n_{2}\mu _{2}$

$s:=n_{1}\sigma _{1}^{2}+n_{2}\sigma _{2}^{2}$

$\limfunc{svar}:=\limfunc{nplaces}\left( 1-\func{NormalDist}\left( a;m,\sqrt{%
s}\right) ,3\right) $

\subsubsection{Statement}

P\aa\ f\"{a}rjor \"{a}r det viktigt att lasten inte blir f\"{o}r stor. P\aa\ %
en viss f\"{a}rjelinje antar man att vikten hos personbilar inklusive last
och passagerare \"{a}r $%
%TCIMACRO{\FORMULA{\mu _{1}}{\mu _{1}}{evaluatenum}}%
%BeginExpansion
\mu _{1}%
%EndExpansion
$ ton i medeltal med standardavvikelsen $%
%TCIMACRO{\FORMULA{\sigma _{1}}{\sigma _{1}}{evaluatenum}}%
%BeginExpansion
\sigma _{1}%
%EndExpansion
$ ton. Lastbilarnas vikter antas vara i medeltal $%
%TCIMACRO{\FORMULA{\mu _{2}}{\mu _{2}}{evaluatenum}}%
%BeginExpansion
\mu _{2}%
%EndExpansion
$ ton med en standardavvikelse p\aa\ $%
%TCIMACRO{\FORMULA{\sigma _{2}}{\sigma _{2}}{evaluatenum}}%
%BeginExpansion
\sigma _{2}%
%EndExpansion
$ ton. P\aa\ en viss resa har $%
%TCIMACRO{\FORMULA{n_{1}}{n_{1}}{evaluatenum}}%
%BeginExpansion
n_{1}%
%EndExpansion
$ personbilar och $%
%TCIMACRO{\FORMULA{n_{2}}{n_{2}}{evaluatenum}}%
%BeginExpansion
n_{2}%
%EndExpansion
$ lastbilar bokat plats. Alla fordons vikter \"{a}r oberoende och normalf%
\"{o}rdelade. Ber\"{a}kna, med tre decimalers noggranhet, sannolikheten att
bilarnas sammanlagda vikt \"{o}verstiger $%
%TCIMACRO{\FORMULA{a}{a}{evaluatenum}}%
%BeginExpansion
a%
%EndExpansion
$ ton.

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}$)

\subsubsection{Solution}

S\"{a}tt 
\begin{align*}
L_{i}& =\text{lastbil }i\text{:s vikt}\in F\left( 
%TCIMACRO{\FORMULA{\mu _{2}}{\mu _{2}}{evaluatenum}}%
%BeginExpansion
\mu _{2}%
%EndExpansion
,%
%TCIMACRO{\FORMULA{\sigma _{2}}{\sigma _{2}}{evaluatenum}}%
%BeginExpansion
\sigma _{2}%
%EndExpansion
\right) \\
P_{j}& =\text{personbil }j\text{:s vikt}\in F\left( 
%TCIMACRO{\FORMULA{\mu _{1}}{\mu _{1}}{evaluatenum}}%
%BeginExpansion
\mu _{1}%
%EndExpansion
,%
%TCIMACRO{\FORMULA{\sigma _{1}}{\sigma _{1}}{evaluatenum}}%
%BeginExpansion
\sigma _{1}%
%EndExpansion
\right)
\end{align*}%
d\aa\ g\"{a}ller enligt Centrala Gr\"{a}nsv\"{a}rdesatsen att total vikt vid
resan blir (obs medeltal) 
\begin{equation*}
V=\sum_{i=1}^{25}L_{i}+\sum_{j=1}^{36}P_{j}\approx N\left( 
%TCIMACRO{\FORMULA{n_{2}}{n_{2}}{evaluatenum}}%
%BeginExpansion
n_{2}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{\mu _{2}}{\mu _{2}}{evaluatenum}}%
%BeginExpansion
\mu _{2}%
%EndExpansion
+%
%TCIMACRO{\FORMULA{n_{1}}{n_{1}}{evaluatenum}}%
%BeginExpansion
n_{1}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{\mu _{1}}{\mu _{1}}{evaluatenum}}%
%BeginExpansion
\mu _{1}%
%EndExpansion
,\sqrt{%
%TCIMACRO{\FORMULA{n_{2}}{n_{2}}{evaluatenum}}%
%BeginExpansion
n_{2}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{\sigma _{2}}{\sigma _{2}}{evaluatenum}}%
%BeginExpansion
\sigma _{2}%
%EndExpansion
^{2}+%
%TCIMACRO{\FORMULA{n_{1}}{n_{1}}{evaluatenum}}%
%BeginExpansion
n_{1}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{\sigma _{1}}{\sigma _{1}}{evaluatenum}}%
%BeginExpansion
\sigma _{1}%
%EndExpansion
^{2}}\right)
\end{equation*}%
ty oberoende och normalf\"{o}rdelade variabler. Vi erh\aa ller nu den s\"{o}%
kta sannolikheten till 
\begin{equation*}
P\left( V>%
%TCIMACRO{\FORMULA{a}{a}{evaluatenum}}%
%BeginExpansion
a%
%EndExpansion
\right) =1-\func{NormalDist}\left( 
%TCIMACRO{\FORMULA{a}{a}{evaluatenum}}%
%BeginExpansion
a%
%EndExpansion
;%
%TCIMACRO{\FORMULA{m}{m}{evaluatenum}}%
%BeginExpansion
m%
%EndExpansion
,\sqrt{%
%TCIMACRO{\FORMULA{s}{s}{evaluatenum}}%
%BeginExpansion
s%
%EndExpansion
}\right) =%
%TCIMACRO{\FORMULA{\limfunc{svar}}{\limfunc{svar}}{evaluatenum}}%
%BeginExpansion
\limfunc{svar}%
%EndExpansion
\end{equation*}

\end{document}