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\begin{center}   {\Large \textbf{STA 256f18 Assignment Seven}}\footnote{Copyright information is at the end of the last page.}\vspace{1 mm}\end{center}\noindent
Please look at Sections 3.4 through 3.7 in Chapter 3 of the textbook and look over your lecture notes. In the text, don't worry about Bayesian inference and note that in Section 3.7, we are only covering the density of the maximum and minimim, not order statistics in general.

These 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 Three. 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.
\vspace{5mm}
\begin{enumerate} %%%%%%%%%%%%%%%%%%%% Continuous RVs in general %%%%%%%%%%%%%%%%%%%%


\item Do Problem 14 in the text.  

\item Do Problem 18 in the text. 

\item Let $X$ and $Y$ be continuous random variables. 
    \begin{enumerate}
        \item Prove that if $f_{xy}(x,y) = f_x(x) \, f_y(y)$ for all real $x$ and $y$, then the random variables $X$ and $Y$ are independent. This result is also true if the condition holds except on a set of probability zero.
        \item Prove that if $X$ and $Y$ are independent, then $f_{xy}(x,y) = f_x(x) \, f_y(y)$ at all points where $F_{xy}(x,y)$ is differentiable and $f_{xy}(x,y)$ is continuous.
    \end{enumerate}

\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.

\item Let $p_{xy}(x,y) = \frac{xy}{36}$ for $x=1,2,3$ and $y=1,2,3$, and zero otherwise.  
        \begin{enumerate}
            \item What is $p_{y|x}(1|2)$? 
            \item What is $p_{x|y}(1|2)$?
            \item Are $x$ and $y$ independent? Answer Yes or No and prove your answer.
        \end{enumerate}


\item Do Problem 20 in the text. Sketch the support first. % Thomas

\item Problem 23 in the text is too challenging for a test or exam, but if you're up for it, use the Law of Total Probability, watch the limits of summation on n, change the variable of summation using k = n-x, and then apply the Binomial Theorem. 
% Thomas, uncon prob

% \item Do Problem 24 in the text. Thomas suggested this, but it requires conditional of discrete given continuous and vice versa. We have not developed the concepts, though the text has via Bayesian inference.

% T suggests 27 but it's not quite true as stated. My problem above is better.

% T suggests 31 but that asks for an intuitive argument, and it's not a type of question we would ask.

\item Do Problem 50 in the text. 

%%%%%%%%%%%%%%%% Convolutions %%%%%%%%%%%%%%%%

\item Let $X$ and $Y$ be independent discrete random variables. Derive the convolution formula for the probability mass function of $Z = X + Y$. Use the Law of Total Probability.

\item Let $X$ and $Y$ be independent continuous random variables. Derive the convolution formula for the probability density function of $Z = X + Y$. Use Fubini's Theorem, which says you can always switch order of integration if what you are integrating is positive.

\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.

\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.

\item Do Problem 43 in the text. Use the convolution formula. Consider the cases $0 < z \leq 1$ and $1 < z \leq 2$ separately. You need to pay close attention to the limits of integration, which are different for the two cases. If $f(x)$ is the density of the uniform $(0,1)$ distribution, for what values of $x$ is $f(z-x)$ non-zero? You will see that the density of $X = X+Y$ is triangular. % Sum of uniforms. Recommended by Thomas.

\item Let $X$ and $Y$ be independent Gamma random variables with parameters $\alpha$, and $\lambda=1$. Find the probability density function of $Z=X+Y$ and identify it by name. 

Once you apply the convolution formula, you are looking at a difficult integral. What you are trying to integrate looks a bit like a beta density; see the formula sheet. The change of variables $u = \frac{x}{z}$ gets you closer. Multiply and divide by the right quantity and you have it.

%%%%%%%%%%%%%%%% Jacobian %%%%%%%%%%%%%%%%

\item Do Problem 48 in the text, except let $\lambda_1$ and $\lambda_2$ have a single value $\lambda$.  Use the Jacobian method. Identify the distribution of $T_1+T_2$ by name. % Independent exponentials; I can't do it with lambda1 neq lambda2 as in the textbook problem.

 \item Let $X_1$ and $X_2$ be independent standard normal random variables. Find the probability density function of $Y_1 = X_1/X_2$.
 
\item \label{42a} Do Problem 42a in the text. Use the Law of Total Probability and differentiate. % Double exponential with W. 

\item Here is another way to obtain the double exponential density of Question~\ref{42a}. Let $X_1$ and $X_2$ be independent exponential random variables with parameter $\lambda$, and let $Y_1 = X_1 - X_2$. Find the (marginal) probability density of $Y_1$. It's essential and a bit challenging to sketch the region where the joint density of $Y_1$ and $Y_2$ is non-zero.

\item Show that the normal density integrates to one. The formula $dx \, dy = r \, dr \, d\theta$ will be on the formula sheet.

\item Do Problem 61 in the text. 

\item Do Problem 62 in the text. % P(X^2 + Y^2 < 1)

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

\item Let $X_1, \ldots, X_n$ be independent Uniform ($0,\theta$) random variables, and let $Y = \max(X_1, \ldots, X_n)$. Find the density $f_y(y)$.

\end{enumerate}

\vspace{2mm}\noindent
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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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% Promising Ch. 3 problems not assigned: 51, 55, 56, 70, 


\item
    \begin{enumerate}
        \item 
        \item 
    \end{enumerate}


\item Do Problem  in the text. 
\item Do Problem  in the text. 

\item Do Problem  in the text. 


\item
    \begin{enumerate}
        \item 
        \item 
    \end{enumerate}



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