% Sample Question document for STA256 \documentclass[12pt]{article} %\usepackage{amsbsy} % for \boldsymbol and \pmb %\usepackage{graphicx} % To include pdf files! \usepackage{amsmath} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage[colorlinks=true, pdfstartview=FitV, linkcolor=blue, citecolor=blue, urlcolor=blue]{hyperref} % For links \usepackage{fullpage} %\pagestyle{empty} % No page numbers \begin{document} %\enlargethispage*{1000 pt} \begin{center} {\Large \textbf{Sample Questions: Joint Distributions Part One}} STA256 Fall 2019. Copyright information is at the end of the last page. %\rule{6in}{.01in} % Width and height \rule{6in}{.005in} % Horizontal line (Width and height) % \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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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}