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\begin{center}   {\Large \textbf{STA 256f18 Assignment Eight}}\footnote{Copyright information is at the end of the last page.}\vspace{1 mm}\end{center}\noindent
Please read Sections 4.1 and 4.2 in the text, except skip 4.2.1 on measurement error.

The following homework problems are not to be handed in. They are preparation for Term Test 3 and the final exam.  All textbook problems are from Chapter Four. Use the formula sheet to do the problems. On tests and the final exam, you may use anything on the formula sheet unless you are being directly asked to prove it.
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\begin{enumerate} %%%%%%%%%%%%%%%%%%%% Continuous RVs in general %%%%%%%%%%%%%%%%%%%%

\item Let $X$ have a Bernoulli distribution, meaning $P(X=1)=p$ and $P(X=0) = 1-p$. Calculate $E(X)$ and $Var(X)$.

\item Show that if $P(X \geq 0)=1$, then $E(X)\geq 0$. Treat the discrete and continuous cases separately. 

\item Do Problem 5 in the text.  

\item Do Problem 1 in the text. 

\item Let $X$ and $Y$ be independent discrete random variables. Show $E[g(X)h(Y)]=E[g(X)]E[h(Y)]$. Clearly specify where you use independence.

\item Let $X$ and $Y$ be discrete random variables, not necessarily independent. Show $E(aX+bY) = aE(X)+bE(Y)$. You are proving a special case of $E\left(\sum_{i=1}^n a_iX_i \right) = \sum_{i=1}^n a_iE(X_i)$ from the formula sheet, so you can't use that. 

\item Show $Var(X) = E(X^2)-[E(X)]^2$.
\item Prove $Var(a+X) = Var(X)$.
\item Prove $Var(bX) = b^2Var(X)$

\item Let $X$ have a continuous uniform distribution on $(a,b)$. Calculate $E(X)$ and $Var(X)$.
\item Let $X \sim$ Poisson($\lambda$). Calculate $E(X)$ and $Var(X)$. For the variance, it helps to start with $E[X(X-1)]$.
\item Let $X$ have a binomial distribution with parameters $n$ and $p$. Calculate $E(X)$.
\item Let $X$ have a Gamma distribution with parameters $\alpha$ and $\lambda$. Calculate $E(X)$ and $Var(X)$.
\item Let $X \sim N(\mu,\sigma)$. Calculate $E(X)$.
\item Do Problem 8 in the text. 
\item Do Problem 12 in the text. Show $E(X-\xi)=0$. Symmetry means $f(\xi-x)=f(\xi+x)$. Split the integral at $\xi$ and change variables.
\item Do Problem 13 in the text. This is a double integral. Sketch the region of integration and switch order of integration. 
\item Do Problem 16 in the text. 
\item Do Problem 21 in the text. 
\item Do Problem 31 in the text. 
\item Markov's inequality says  that if $P(Y \geq 0)=1$, then $P(Y \geq t) \leq E(Y)/t$. Prove it for the case where $Y$ is a discrete random variable.
\item Chebyshev's inequality says if $X$ is a random variable with $E(X)=\mu$ and $Var(X)=\sigma^2$, then for any $k>0$, $P(|X-\mu| \geq k\sigma) \leq \frac{1}{k^2}$. Use Markov's inequality to prove it. 

\end{enumerate} % End of questions
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This assignment was prepared by 
%\href{https://www.utm.utoronto.ca/math-cs-stats/faculty-staff/zou-dr-nan}{Nan Zou} and 
\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/256f18} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/256f18}}\end{center}

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