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{\Large \textbf{Sample Questions: Expected Value, Variance and Covariance}}
STA256 Fall 2019. Copyright information is at the end of the last page.
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\begin{enumerate}
\item {\Large Let $X$ have a continuous uniform distribution on $(L,R)$. Calculate $E(X)$. }
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\item Recall that a fair game is one with expected value zero. You wager one dollar, and toss a coin with $P(\mbox{Head}) = \theta$. If it's heads, you win. In dollars, what should the payoff be so that the game is fair?
% (1-theta)*(-1) + theta*a = 0 <=> theta*a = 1=theta <=> a = (1-theta)/theta
% So for example, if P(H)=1/100, payout should be $99.
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\item {\Large Let $X \sim$ Poisson($\lambda$). Calculate $E(X)$. }
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\item Let the continuous random variable $X$ have density
{\Large
$f_{_X}(x) = \left\{ \begin{array}{ll}
\frac{\alpha}{x^{\alpha+1}} & \mbox{ for } x \geq 1 \\
0 & \mbox{otherwise}
\end{array} \right. $
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Where $\alpha>0$.
\begin{enumerate}
\item Verify that $f_{_X}(x)$ integrates to one. \vspace{80mm}
\item Calculate $E(X)$. For what values of $\alpha$ does $E(X)$ exist?
\end{enumerate}
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\item {\Large Let $X \sim N(\mu,\sigma^2)$. Calculate $E(X)$. }
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% \item {\Large Let $X$ have a binomial distribution with parameters $n$ and $p$. Calculate $E(X)$. }
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\item {\Large Let $X$ have a Gamma distribution with parameters $\alpha$ and $\lambda$. Calculate $E(X^k)$. } % Update!
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% \item {\Large Prove $Var(a+X) = Var(X)$. }
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\item {\Large Prove $Var(bX) = b^2Var(X)$. }
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\item {\Large Show $Var(X) = E(X^2)-[E(X)]^2$.}
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%\item {\Large Let $X \sim$ Uniform(0,1). Calculate $Var(X)$.}
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\item {\Large Let $X$ have density $e^{-x}$ for $x \geq 0$ and zero otherwise. Calculate $Var(X)$.}
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\item {\Large Let $X \sim N(\mu,\sigma^2)$. Calculate $Var(X)$.}
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\item {\Large Show $Cov(X,Y) = E(XY) - E(X) E(Y)$.}
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\item {\Large Let $X$ and $Y$ be independent (continuous) random variables. Show $E(XY)=E(X)E(Y)$. }
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\item If $X$ and $Y$ are independent, $Cov(X,Y) = $
\end{enumerate} % End of all the questions
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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 {\Large The discrete random variables $x$ and $y$ have joint distribution
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\begin{tabular}{c|ccc}
& $x=1$ & $x=2$ & $x=3$ \\ \hline
$y=1$ & $3/12$ & $1/12$ & $3/12$ \\
$y=2$ & $1/12$ & $3/12$ & $1/12$ \\
\end{tabular}
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\begin{enumerate}
\item What is $E(X|Y=1)$? \vspace{80mm}
\item What is $E(Y^2|X=2)$? \vspace{70mm}
\end{enumerate}
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\begin{tabular}{c|ccc}
& $x=1$ & $x=2$ & $x=3$ \\ \hline
$y=1$ & $3/12$ & $1/12$ & $3/12$ \\
$y=2$ & $1/12$ & $3/12$ & $1/12$ \\
\end{tabular}
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\item % Simple but requiring a clear head. Of course E(Y|X=x) has to equal x^2/2 unless you make a mistake.
{\large
Let $f_{x,y}(x,y) = 3$ for $0 < x < 1 $ and $ 0 < y < x^2$, and zero otherwise. \vspace{40mm}
\begin{enumerate}
\item Using $f_x(x)=3x^2$ for $0y$ separately.
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\item {\Large }
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More questions, maybe for HW.
\begin{enumerate}
\item Let $X$ and $Y$ be independent (discrete) random variables. Show $E\left(g(X)h(Y)\right)=E\left(g(X)\right)E\left(h(Y)\right)$.
\item Let $X \sim$ Uniform(1,2). Find $E(\frac{1}{X})$.
\item True or false? $E(\frac{1}{X}) = 1/E(X)$. If it is true, prove it in general. If it is false, give a counter-example.
\item
\item
\end{enumerate}