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\begin{center}   
{\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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% \vspace{3 mm}
\end{center}
\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
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    \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
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    \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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    \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
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    \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
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    \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
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    \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 $x<y$ and $x>y$ 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
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    \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
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    \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
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    \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}













