% 256f19Assignment6.tex             Continuous distributions
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{\Large \textbf{\href{http://www.utstat.toronto.edu/~brunner/oldclass/256f19}{STA 256f19} 
Assignment Six}}\footnote{Copyright information is at the end of the last page.}
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\noindent
Please read Sections 2.5 and 2.7. Also, look over your lecture notes. The following homework problems are not to be handed in. They are preparation for Term Test 2 and the final exam. Use the formula sheet.

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\begin{enumerate} 

% CDFs

\item Let $X$ have an exponential density with parameter $\lambda>0$. Find the cumulative distribution function $F_{_X}(x)$.

\item Let $X$ have a Normal($\mu,\sigma^2$) density. Express the cumulative distribution function $F_{_X}(x)$ in terms of $\Phi(z)$, the cumulative distribution function of a Normal random variable with $\mu=0$ and $\sigma^2=1$. The notation $\Phi$ is standard.

\item Do exercises 2.5.7, 2.5.9, 2.5.21, 2.5.23 in the text. 

\item Let $X$ be a random variable with cumulative distribution function $F_{_X}(x)$
    \begin{enumerate}
        \item Prove $\displaystyle \lim_{x \rightarrow \infty} F_{_X}(x) = 1$.
        \item Prove $\displaystyle \lim_{x \rightarrow -\infty} F_{_X}(x) = 0$.
    \end{enumerate}
% A few more one-variable transformations
\item Do Exercises 2.6.1, 2.6.3, 2.6.4, 2.6.5, 2.6.7 and 2.6.8 in the text. Please use the ``distribution function technique" illustrated in lecture, and \emph{not} theorems from Section 2.6. In particular, Theorems 2.6.1-2.6.4 are to be avoided and will not be on the formula sheet.

%%%%%%%%%%%%% Joint distributions part one %%%%%%%%%%%%%

\item Let $p_{_{X,Y}}(x,y) = c(x+y)$ for $x=1,2,3$, $y = 1,2$, and zero otherwise. 
    \begin{enumerate}
        \item Find the constant $c$. [21]
        \item What is $p_{_{X,Y}}(1,1)$?   [2/21]
        \item What is $p_{_{X,Y}}(2.5,1.75)$?   [0]
        \item What is $F_{_{X,Y}}(2.5,1.75)$?   [5/21]
        \item What is $F_{_{X,Y}}(5,1.5)$?   [9/21]
        \item What is $F_{_{X,Y}}(0,4)$?   [0]
        \item What is $F_{_{X,Y}}(4,4)$?   [1]
        \item What is $p_{_X}(2)$?   [7/21]
        \item What is $p_{_Y}(1)$?   [9/21]
        \item What is $F_{_X}(2.5)$? [12/21]
    \end{enumerate}

\item Do Exercise 2.7.1 in the text. Don't waste too much energy trying to understand the answer in the back of the  book. Instead of giving the joint cumulative distribution function, answer the following questions instead. Notice that you are \emph{not} being asked for a full statement of the cumulative distribution function. 
    \begin{enumerate}
        \item Draw a set of $x,y$ coordinates, plot all the points with non-zero probability, and write a probability beside each point.
        \item What is $p_{_{X,Y}}(x,y)$? Make sure your answer applies to all real $x$ and $y$.
        \item What is $F_{_{X,Y}}(1,-1)$? I hope you agree it's $P(X=0,Y=-2) = \frac{2}{3}$. Thus, the answer in the back of the book has to be wrong.
    \end{enumerate}
    
\item Do Exercise 2.7.3 in the text. It's easier if you put the joint probabilities in a two-way table.

\item Do Exercise 2.7.5 in the text. Hint: Think of sets contained in other sets.

\item Five cards are selected from an ordinary deck of 52 playing cards. Let $X$ equal the number of spades and $Y$ equal the number of hearts. Give the joint probability function of $X$ and $Y$
    \begin{enumerate}
        \item If the sampling is \emph{without} replacement. Be sure to specify the values of $x$ and $y$ for which $p_{_{X,Y}}(x,y)$ is non-zero. 
        \item If the sampling is \emph{with} replacement. Be sure to specify the values of $x$ and $y$ for which $p_{_{X,Y}}(x,y)$ is non-zero
    \end{enumerate}

\item Please look at Example 2.7.3 on page 82 of the text. 
    \begin{enumerate}
        \item Find the joint density $f_{_{X,Y}}(x,y)$. Show your work. Make sure to specify where it is non-zero.
        \item Check that your $f_{_{X,Y}}(x,y)$ integrates to one.
        \item Find the marginal density $f_{_X}(x)$. If you do not specify where the density is non-zero, it's worth half marks at most.
        \item Find the marginal density $f_{_Y}(y)$. If you do not specify where the density is non-zero, it's worth half marks at most.
        \item Find the marginal cumulative distribution function $F_{_X}(x)$.
                \begin{enumerate}
                    \item By integrating $f_{_X}(x)$. 
                    \item By taking limits of $F_{_{X,Y}}(x,y)$
                \end{enumerate}
        Make sure your answer applies to all real $x$.

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        \item Find the marginal cumulative distribution function $F_{_Y}(y)$.
                \begin{enumerate}
                    \item By integrating $f_{_Y}(y)$. 
                    \item By taking limits of $F_{_{X,Y}}(x,y)$
                \end{enumerate}
        Make sure your answer applies to all real $y$.
        \item Find $P(Y < X^2)$. Show your work. My answer is $\frac{1}{5}$.
    \end{enumerate}

\item Do Exercise 2.7.9 in the text. Hint: Sketch the support before beginning. ][% Marginals with non-rectangular support. 

\item Do Problem 2.7.12 in the text. Hint: Use Exercise 2.7.5.

\item Do Problem 2.7.16 in the text. % Like one of my sample problems, by chance.



\end{enumerate} % End of all the questions

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\noindent
This assignment 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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