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\begin{center}   {\Large \textbf{STA 256f18 Assignment Six}}\footnote{Copyright information is at the end of the last page.}\vspace{1 mm}\end{center}\noindent
Please read Sections 3.2 and 3.3 in Chapter 3 of the textbook and look over your lecture notes. These homework problems are not to be handed in. They are preparation for Term Test 2 and the final exam.  All textbook problems are from Chapter Three. Use the formula sheet to do the problems. You will have a copy of Table~2 from Appendix~B in the text. 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 \label{Discretetable} Do Problem 1a in the text. 

\item For the joint distribution of Problem~\ref{Discretetable}, give
    \begin{enumerate}
        \item $F_{xy}(3,2)$ The answer is a number. % 0.47
        \item $F_{xy}(4,4)$ The answer is a number. % 1
        \item $F_{xy}(0,0)$ The answer is a number. % 0
        \item $F_{xy}(2,1.5)$ The answer is a number. % 0.51
        \item $F_{xy}(3.5,7)$ The answer is a number. % 1 - 0.18 = 0.82
        \item $F_x(x)$. Your answer must apply to all real $x$. % 
    \end{enumerate}

\item Let $p_{xy}(x,y) = c(x+y)$ for $x=1,2,3$, $y = 1,2$, and zero otherwise. 
    \begin{enumerate}
        \item Find the constant $c$. The answer is a number.
        \item What is $p_x(2)$? The answer is a number.
        \item What is $p_y(1)$? The answer is a number.
        \item What is $F_x(2.5)$? The answer is a number.
    \end{enumerate}

\item Do Problem 2 in the text. % Multivariate hypergeometric

\item Do Problem 3 in the text. % Multinomial

\item Do Problem 6 in the text. Assume $a>0$ and $b>0$. It helps to sketch the ellipse. The area of the ellipse is $\pi ab$, a fact you may use without proof. If you are interested in the derivation of the formula for area, note that $2\int_{-a}^a \sqrt{a^2+x^2} \, dx$ is the area of a circle with radius $a$.

\item Do Problem 7 in the text. % CDF of ind. exponentials.
\pagebreak

\item Do Problem 8ab in the text.  % Joint density, P(X>Y) etc. and marginals.

\item Do Problem 9a in the text. % Joint density, asks for marginals again.

\item Let $X$ and $Y$ be continuous random variables. Show $\lim_{x \rightarrow \infty} F_{xy}(x,y)) = F_y(y)$. Use Fubini's Theorem, which says you can always switch order of integration as long as what you are integrating is non-negative. Don't just move limits through integrals.

    \item The continuous random variables $X$ and $Y$ have joint cumulative distribution function 
\begin{displaymath}
    F_{xy}(x,y) = \left\{ \begin{array}{ll} % ll means left left
   x^2 (1 - e^{-3y}) & \mbox{for $ 0 \leq x \leq 1$ and $y \geq 0$}  \\
   1-e^{-3y}         & \mbox{for $ x > 1$ and $y \geq 0$}  \\
   0     & \mbox{otherwise}  
   \end{array}  \right. % Need that crazy invisible right period!
\end{displaymath}
        \begin{enumerate}
            \item What is $F_{xy}(\frac{1}{2},3)$? 
            \item What is $F_{xy}(2,1)$? 
            \item What is $F_{xy}(-1,3)$? 
            \item What is $f_{xy}(x,y)$?
            \item Obtain $f_y(y)$ by integrating out $x$.
            \item Obtain $F_y(y)$ by taking limits.
            \item Obtain $f_y(y)$ by differentiation.
        \end{enumerate}

\item Do Problem 10ab in the text. This problem requires too much geometric intuition to appear on a test or exam, but it's a valuable exercise anyway because it requires you to pay attention to limits of integration. The volume of a sphere is $\frac{4}{3}\pi r^3$. Don't hesitate to use your knowledge that the area of a circle is $\pi r^2$. 

\item Do Problem 12ab in the text. 

\item Do Problem 15abc in the text. Make sure you do 10b first. Part (b) of this question says sketch the joint density, but they must mean sketch the region where the joint density is non-zero.

\item Do Problem 17ab in the text. 

\item Do Problem 18abc in the text. In addition, find $P(X+Y>1)$.

\end{enumerate}

\vspace{2mm}\noindent
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\begin{tabular}{l}\hspace{6in} \\ \hline\end{tabular}
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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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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\item %%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%\item     \begin{enumerate}        \item         \item     \end{enumerate}\item \item \item \item Do Problem 8 in the text.%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%\item     \begin{enumerate}        \item         \item     \end{enumerate}