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{\Large \textbf{Sample Questions: Joint Distributions Part Two}}

STA256 Fall 2018. Copyright information is at the end of the last page.
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\begin{enumerate} 

    \item Let $X$ and $Y$ be continuous random variables. Prove that $X$ and $Y$ are independent if and only if $f_{xy}(x,y) = f_x(x) \, f_y(y)$.
    

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    \item Let $X$ and $Y$ be discrete random variables. Prove that if $p_{xy}(x,y) = p_x(x) \, p_y(y)$, then $X$ and $Y$ are independent.


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    \item Let $X$ and $Y$ be discrete random variables. Prove that if $X$ and $Y$ are independent, then 
    $p_{xy}(x,y) = p_x(x) \, p_y(y)$. % Need extra paper.


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    \item Let {\Large $p_{xy}(x,y) = \frac{|x-2y|}{19}$} 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}

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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? Answer Yes or No and prove your answer.
        \end{enumerate}

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    \item Let $X \sim$ Poisson($\lambda_1$) and $Y \sim$ Poisson($\lambda_2$) be independent. Using the convolution formula, find the probability mass function of $Z=X+Y$ and identify it by name.

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    \item Let $X \sim$ Binomial($n_1,p$) and $Y \sim$ Binomial($n_2,p$) be independent. Using the convolution formula, find the probability mass function of $Z=X+Y$ and identify it by name.

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    \item Let $X$ and $Y$ be independent exponential random variables with parameter $\lambda$. Using the convolution formula, find the probability density function of $Z=X+Y$ and identify it by name. 

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    \item Let $X_1$ and $X_2$ be independent standard normal random variables. Find the probability density function of $Y_1 = X_1/X_2$.


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    \item Use the Jacobian method to prove the convolution formula for continuous random variables.


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    \item Show that the normal probability density function integrates to one.


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    \item Prove $\Gamma\left(\frac{1}{2}\right) = \sqrt{\pi}$.  



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    \item Let $X_1, \ldots, X_n$ be independent random variables with probability density function $f(x)$ and cumulative distribution function $F(x)$. Let $Y = \max(X_1, \ldots, X_n)$. Find the density $f_y(y)$. 


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    \item Let $X_1, \ldots, X_n$ be independent random variables with probability density function $f(x) = e^{-x}$ for $x \geq 0$. Let $Y = \max(X_1, \ldots, X_n)$. Find the density $f_y(y)$.


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    \item Let $X_1, \ldots, X_n$ be independent random variables with probability density function $f(x)$ and cumulative distribution function $F(x)$. Let $Y = \min(X_1, \ldots, X_n)$. Find the density $f_y(y)$. 


\end{enumerate}

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\noindent
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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: 
     
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\href{http://www.utstat.toronto.edu/~brunner/oldclass/256f18} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/256f18}}
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    \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 $F_{xy}(x,y)$. Consider $x<y$ and $x>y$ separately.