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{\Large \textbf{Sample Questions: Conditional Distributions and Independent Random Variables}}
STA256 Fall 2019. Copyright information is at the end of the last page.
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\item Let $X$ and $Y$ be continuous random variables. Prove that $X$ and $Y$ are independent if and only if $f_{_{X,Y}}(x,y) = f_{_X}(x) \, f_{_Y}(y)$.
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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.
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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.
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\item Let {\Large $p_{_{X,Y}}(x,y) = \frac{|x-2y|}{20}$} 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. $
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\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 $X_1, \ldots, X_n$ be independent random variables with probability density function $f_{_X}(x)$ and cumulative distribution function $F_{_X}(x)$. 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 $f_{_X}(x) = e^{-x}$ 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 $f_{_X}(x)$ and cumulative distribution function $F_{_X}(x)$. Let $Y = \min(X_1, \ldots, X_n)$. Find the density $f_{_Y}(y)$.
\end{enumerate}
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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/256f19} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/256f19}}
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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 $F_{_{X,Y}}(x,y)$. Consider $xy$ separately.