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{\Large \textbf{Sample Questions: Joint Distributions Part One}}
STA256 Fall 2019. Copyright information is at the end of the last page.
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\begin{enumerate}
\item The discrete random variables $x$ and $y$ have joint probability mass function $p_{_{X,Y}}=cxy$ for $x=1,2,3$, $y = 1,2$, and zero otherwise.
\begin{enumerate}
\item Find the value of the constant $c$ and calculate the marginal probability functions. \vspace{80mm}
\item What is $F_{_X}(x)$?
\end{enumerate}
\pagebreak
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\item The discrete random variables $x$ and $y$ have joint distribution
\begin{center}
\begin{tabular}{c|ccc}
& $x=1$ & $x=2$ & $x=3$ \\ \hline
$y=2$ & $3/12$ & $1/12$ & $3/12$ \\
$y=1$ & $1/12$ & $3/12$ & $1/12$ \\
\end{tabular}
\end{center}
Give the following. The answers are numbers.\vspace{10mm}
\begin{enumerate}
\item $F_{_{X,Y}}(1,1)$ \hspace{60mm}
$F_{_{X,Y}}(2,2)$ \vspace{15mm}
\item $F_{_{X,Y}}(1.5,4)$ \hspace{60mm}
$F_{_{X,Y}}(-1,3)$ \vspace{15mm}
\item $F_{_{X,Y}}(4,4)$ \hspace{60mm}
$F_{_{X,Y}}(6,1.82)$ \vspace{15mm}
\item $F_{_{X,Y}}(4,19)$ \hspace{60mm}
$F_{_{X,Y}}(0,0)$ \vspace{15mm}
\end{enumerate}
\begin{center}
\begin{tabular}{c|ccc}
& $x=1$ & $x=2$ & $x=3$ \\ \hline
$y=2$ & $3/12$ & $1/12$ & $3/12$ \\
$y=1$ & $1/12$ & $3/12$ & $1/12$ \\
\end{tabular}
\end{center}
\pagebreak
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\item A jar contains 30 red marbles, 50 green marbles and 20 blue marbles. A sample of 15 marbles is selected \emph{with replacement}. Let $X$ be the number of red marbles and $Y$ be the number of blue marbles. What is the joint probability mass function of $X$ and $Y$? \vspace{15mm}
$p(x,y) = $ \vspace{40mm}
\item This time the selection is without replacement. Again, what is the joint probability mass function of $X$ and $Y$? \vspace{30mm}
$p(x,y) = $
\pagebreak
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\item Let
{\Large
$f_{_{X,Y}}(x,y) = \left\{ \begin{array}{ll}
c(x+y) & \mbox{for $0 \leq x \leq 1$ and $ 0 \leq y \leq x$ } \\
0 & \mbox{otherwise}
\end{array} \right. $
} % End size
\begin{enumerate}
\item Find the constant $c$. \vspace{100mm}
\item What is $f_{_X}(x)$?
\end{enumerate}
\pagebreak
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\item The continuous random variables $X$ and $Y$ have joint cumulative distribution function
\linebreak \vspace{2mm}
{\Large $F_{_{X,Y}}(x,y) = \left\{ \begin{array}{ll} % ll means left left
x^3 - x^3 e^{-y/4} & \mbox{for $ 0 \leq x \leq 1$ and $y \geq 0$} \\
1-e^{-y/4} & \mbox{for $ x > 1$ and $y \geq 0$} \\
0 & \mbox{otherwise}
\end{array} \right. $ % Need that crazy invisible right period! $
} % End size
\vspace{4mm}
\begin{enumerate}
\item What is $F_{_{X,Y}}(\frac{1}{2},3)$? \vspace{15mm}
\item What is $F_{_{X,Y}}(2,3)$? \vspace{15mm}
\item What is $F_{_{X,Y}}(-1,3)$? \vspace{15mm}
\item What is $f_{_{X,Y}}(x,y)$?
\end{enumerate}
\pagebreak
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\item Still for the joint distribution with
$F_{_{X,Y}}(x,y) = \left\{ \begin{array}{ll} % ll means left left
x^3 - x^3 e^{-y/4} & \mbox{for $ 0 \leq x \leq 1$ and $y \geq 0$} \\
1-e^{-y/4} & \mbox{for $ x > 1$ and $y \geq 0$} \\
0 & \mbox{otherwise}
\end{array} \right. $ and \vspace{2mm}
\noindent
$f_{_{X,Y}}(x,y) = \left\{ \begin{array}{ll}
3x^2 \frac{1}{4}e^{-y/4} & \mbox{for $ 0 \leq x \leq 1$ and $y \geq 0$} \\
0 & \mbox{otherwise}
\end{array} \right. $
\begin{enumerate}
\item Obtain $f_{_X}(x)$ by integrating out $y$. \vspace{70mm}
\item Calculate $F_{_X}(x)$ by taking limits. \vspace{60mm}
\item Obtain $f_{_X}(x)$ from $F_{_X}(x)$.
\pagebreak
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For
$F_{_{X,Y}}(x,y) = \left\{ \begin{array}{ll} % ll means left left
x^3 - x^3 e^{-y/4} & \mbox{for $ 0 \leq x \leq 1$ and $y \geq 0$} \\
1-e^{-y/4} & \mbox{for $ x > 1$ and $y \geq 0$} \\
0 & \mbox{otherwise}
\end{array} \right. $ and \vspace{2mm}
\noindent
$f_{_{X,Y}}(x,y) = \left\{ \begin{array}{ll}
3x^2 \frac{1}{4}e^{-y/4} & \mbox{for $ 0 \leq x \leq 1$ and $y \geq 0$} \\
0 & \mbox{otherwise}
\end{array} \right. $
\item Obtain $f_{_Y}(y)$ by integrating out $x$. \vspace{70mm}
\item Obtain $F_{_Y}(y)$ by taking limits. \vspace{60mm}
\item Obtain $f_{_Y}(y)$ from $F_{_Y}(y)$.
\end{enumerate}
\pagebreak
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\item Let
{\Large
$f_{_{X,Y}}(x,y) = \left\{ \begin{array}{ll}
2e^{-(x+y)} & \mbox{for $ 0 \leq x \leq y$ and $y \geq 0$} \\
0 & \mbox{otherwise}
\end{array} \right. $
\noindent
Obtain $f_{_X}(x)$.
} % End size
\pagebreak
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\item Let
{\Large
$f_{_{X,Y}}(x,y) = \left\{ \begin{array}{ll}
\frac{xy}{16} & \mbox{for $ 0 \leq x \leq 2$ and $0 \leq y \leq 4$} \\
0 & \mbox{otherwise}
\end{array} \right. $
} % End size
\noindent
Find $P(Y < X^2)$. The answer is a number.
\pagebreak
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\item Let
{\Large
$f_{_{X,Y}}(x,y) = \left\{ \begin{array}{ll}
4xy \, e^{-(x^2+y^2)} & \mbox{for $ x \geq 0$ and $y \geq 0$} \\
0 & \mbox{otherwise}
\end{array} \right. $
} % End size
\noindent
Find $P(Y > X)$. The answer is a number.
\end{enumerate}
\vspace{155mm}
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\noindent
\begin{center}\begin{tabular}{l}
\hspace{6in} \\ \hline
\end{tabular}\end{center}
This handout was prepared by \href{http://www.utstat.toronto.edu/~brunner}{Jerry Brunner},
Department of Mathematical and Computational Sciences, University of Toronto. It is licensed under a
\href{http://creativecommons.org/licenses/by-sa/3.0/deed.en_US}
{Creative Commons Attribution - ShareAlike 3.0 Unported License}. Use any part of it as you like and share the result freely. The \LaTeX~source code is available from the course website:
\begin{center}
\href{http://www.utstat.toronto.edu/~brunner/oldclass/256f19} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/256f19}}
\end{center}
\end{document}
% The answer is a number. Circle your answer.
% MESSY!
\item Continuing with
{\Large
$f_{_{X,Y}}(x,y) = \left\{ \begin{array}{ll}
2e^{-(x+y)} & \mbox{for $ 0 \leq x \leq y$ and $y \geq 0$} \\
0 & \mbox{otherwise}
\end{array} \right. $
\noindent
Obtain $F_{_{X,Y}}(x,y)$. Consider $xy$ separately.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\pagebreak
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\item Let $X$ and $Y$ be continuous random variables. Prove that $X$ and $Y$ are independent if and only if $f_{_{X,Y}}(x,y) = f_{_X}(x) \, f_{_Y}(y)$.
\pagebreak
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\item Let $X$ and $Y$ be discrete random variables. Prove that if $p_{_{X,Y}}(x,y) = p_{_X}(x) \, p_{_Y}(y)$, then $X$ and $Y$ are independent.
\pagebreak
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\item Let $X$ and $Y$ be discrete random variables. Prove that if $X$ and $Y$ are independent, then
$p_{_{X,Y}}(x,y) = p_{_X}(x) \, p_{_Y}(y)$. % Need extra paper.
\pagebreak
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\item Let {\Large $p_{_{X,Y}}(x,y) = \frac{|x-2y|}{19}$} for $x=1,2,3$ and $y=1,2,3$, and zero otherwise. \vspace{80mm}
\begin{enumerate}
\item What is {\Large$p_{y|x}(1|2)$}? \vspace{30mm} % 0/5 = 0
\item What is {\Large$p_{x|y}(1|2)$}? \vspace{30mm} % 3/6 = 1/2
\item Are $x$ and $y$ independent? Answer Yes or No and prove your answer.
\end{enumerate}
\pagebreak
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\item Let
{\Large
$f_{_{X,Y}}(x,y) = \left\{ \begin{array}{ll}
2e^{-(x+y)} & \mbox{for $ 0 \leq x \leq y$ and $y \geq 0$} \\
0 & \mbox{otherwise}
\end{array} \right. $
} % End size
\begin{enumerate}
\item Find $f_{x|y}(x|y)$. \vspace{120mm}
\item Are $X$ and $Y$ independent? Prove your answer.
\end{enumerate}