% 256f19Assignment4.tex             Discrete distributions
\documentclass[11pt]{article} 
%\usepackage{amsbsy} % for \boldsymbol and \pmb 
%\usepackage{graphicx} % To include pdf files!
\usepackage{amsmath}
\usepackage{amsbsy}
\usepackage{amsfonts}
\usepackage[colorlinks=true, pdfstartview=FitV, linkcolor=blue, citecolor=blue, urlcolor=blue]{hyperref} % For links
\usepackage{fullpage}
% \pagestyle{empty} % No page numbers


\begin{document}
%\enlargethispage*{1000 pt} 

\begin{center}   
{\Large \textbf{\href{http://www.utstat.toronto.edu/~brunner/oldclass/256f19}{STA 256f19} 
Assignment Four}}\footnote{Copyright information is at the end of the last page.}
\vspace{1 mm}
\end{center}

\noindent
Please read Sections 2.1-2.3 (pages 34-53) in the text, 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.  

%\vspace{5mm}
\begin{enumerate} 

\item Do exercises 2.1.1 and 2.1.9. For 2.1.9, see Example 2.1.6 on indicator functions.

\item Roll two fair dice, and let $X$ be the minimum of the two numbers showing. 
    \begin{enumerate}
        \item Give $p(x)$ and $F(x)$ for $x = 1, \ldots, 6$. % 11,9,7,5,3,1 / 36

\begin{tabular}{ccccccc}
$x$     & 1     & 2    & 3    & 4    & 5    & 6 \\
$p(x)$  & 11/36 & 9/36 & 7/36 & 5/36 & 3/36 & 1/36 \\
$F(x)$  & 11/36 & 20/36 & 27/36 & 32/36 & 35/36 & 36/36 
\end{tabular}

        \item What is $F(1)$? (11/36)
        \item What is $F(3)$? (27/36)
        \item What is $F(3.5)$? (27/36)
        \item What is $p(3.5)$? (0)
        \item What is $F(0)$? (0)
        \item What is $p(0)$? (0)
        \item What is $F(-14)$? (0)
        \item What is $F(14)$? (1)
        \item What is $p(14)$? (0)
    \end{enumerate}

\item Let $p_{_X}(x) = c x$ for $x = 1,2,3$ and zero otherwise.
    \begin{enumerate}
        \item What is the constant $c$? (1/6)
        \item Graph the cumulative distribution function $F_{_X}(x)$. Don't forget right continuity.
        \item \label{fulldist} Write the full probability distribution of $X$ using indicator functions as in Examples 2.2.2 and 2.2.3 in the text. For any set $B \in \mathbb{R}$, $P(X \in B) = \ldots$ 
    \end{enumerate}

\item Do exercise 2.2.7 in the text. 

\item Do exercise 2.2.5 in the text. Assume the chips are mixed thoroughly, so that $X$ and $Y$ are independent.

\item Let the discrete random variable $X$ have probability mass function $p(x) = c \, x^2$ for $x=-2, -1, 0, 1, 2$ and zero otherwise. What is the constant $c$? (1/10)

\item Let the discrete random variable $X$ have probability mass function $p(x) = c \, \frac{2^x}{x!}$ for $x=0,1, \ldots$ and zero otherwise. What is the constant $c$? ($e^{-2}$)

\item Do exercise 2.3.1 in the text. (See the Sample Questions).

\item Do exercise 2.3.3 in the text. Also, what is the cumulative distribution function $F(x)$? Make sure your answer applies to all real $x$. Graph the cumulative distribution function $F(x)$.  Don't forget right continuity.

\item Do exercise 2.3.7 in the text. Hint: Differentiate $\ln(p(x))$.

\item Show that the Bernoulli probabilities sum to one. Use the formula sheet. 
\item Show that the Binomial probabilities sum to one. Use the formula sheet. 
\item Show that the Geometric probabilities sum to one. Use the formula sheet. 
\item Show that the Poisson probabilities sum to one. Use the formula sheet. 

\item Let $X$ be a geometric random variable with parameter $\theta$. 
    \begin{enumerate}
        \item Find $P(X\geq x)$ (Answer is $(1-\theta)^x$)
        \item Find $P(X\geq x+k|X\geq k)$ (Answer is $(1-\theta)^x$)
    \end{enumerate}
This is called the ``memoryless" property of the geometric distribution. It makes sense in terms of coin tossing. % , because ``the coin has no memory."

\item Do exercise 2.3.15 in the text. 

\item Let $X$ be a binomial ($n,\theta$) random variable. Let $n \rightarrow \infty$ and $\theta \rightarrow 0$ in such a way that the value of $n\theta=\lambda$ remains fixed. Show that the probability mass function of $X$ approaches the probability mass function of a Poisson as $n \rightarrow \infty$. This is called the Poisson approximation to the binomial. (See the Sample Questions.)

\item Do exercise 2.3.19 in the text. 

\end{enumerate}

\vspace{20mm} \hrule
\vspace{3mm}

\noindent
The answer to Question \ref{fulldist} is 
$P(X \in B) = \frac{1}{6}I_B(1) +  \frac{2}{6}I_B(2) +  \frac{3}{6}I_B(3)$.

\vspace{3mm} \hrule

\vspace{20mm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{3mm}
\hrule
\vspace{3mm}

\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: 
     
\begin{center}
\href{http://www.utstat.toronto.edu/~brunner/oldclass/256f19} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/256f19}}
\end{center}


\end{document}




\href{http://www.utstat.toronto.edu/~brunner/oldclass/256f18}{STA 256f18}
