% 256f19Assignment9.tex             Moment-generating functions
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{\Large \textbf{\href{http://www.utstat.toronto.edu/~brunner/oldclass/256f19}{STA 256f19} 
Assignment Nine}}\footnote{Copyright information is at the end of the last page.}
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\noindent
Please read Section 3.4 in the text.  Also, look over your lecture notes. The following homework problems are not to be handed in. They are preparation for Term Test 3 and the final exam. Use the formula sheet.


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\begin{enumerate} 

%%%%%%%% Derive, generate moments

\item \label{j1} Let $X$ have a moment-generating function $M_{_X}(t)$ and let $a$ be a constant. Show $M_{_{aX}}(t) = M_{_X}(at)$.

\item Let $X$ have a moment-generating function $M_{_X}(t)$ and let $a$ be a constant. Show $M_{_{a+X}}(t) = e^{at}M_{_X}(t)$.

\item  Let $X_1$ and $X_2$ be independent, discrete random variables, and let $Y = g(X_1)+h(X_2)$.  Show $M_{_Y}(t) = M_{_{g(X_1)}}(t) \, M_{_{h(X_2)}}(t)$. Because the random variables are discrete, you will add rather than integrating.

\item In the following table, derive the moment-generating functions (given on the formula sheet), and then use them to obtain the expected values and variances. To make the task shorter, notice that the Bernoulli is a special case of the binomial, and that the exponential and chi-squared distributions are special cases of the gamma. Chi-squared is a gamma with $\alpha=\nu/2$ and $\lambda = \frac{1}{2}$; exponential is a gamma with $\alpha=1$.
Do the general cases first and then just write the answer for the special cases.

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\begin{tabular}{|l|c|c|c|} \hline
\textbf{Distribution} & \textbf{MGF} $M_x(t)$ & $E(X)$ & $Var(X)$ \\ \hline
Bernoulli ($\theta$)  &  &  & \\ \hline 
Binomial ($n,\theta$) &  &  & \\ \hline 
%Geometric ($p$) &  &  & \\ \hline 
Poisson ($\lambda$) &  &  & \\ \hline 
Exponential ($\lambda$) &  &  & \\ \hline 
Gamma ($\alpha,\lambda$) &   &  & \\ \hline 
Normal ($\mu,\sigma^2$) &  &  & \\ \hline 
Chi-squared ($\nu$) &  &  & \\ \hline 
\end{tabular}
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\item Let $X$ be a geometric random variable with parameter $\theta$. 
    \begin{enumerate}
        \item Find the moment-generating function.
        \item Differentiate to obtain $E(X)$.
    \end{enumerate}


%%%%%%%% Identify to get distribution

\item Let $X \sim N(\mu,\sigma^2)$. Show $Z = \frac{X-\mu}{\sigma} \sim N(0,1)$ using moment-generating functions.

\item Let $X_1 \sim N(\mu_1,\sigma^2_1)$ and $X_2 \sim N(\mu_2,\sigma^2_2)$ be independent. Find the distribution of  $Y = X_1+3X_2$. 

\item Let $X_1, \ldots, X_n$ be independent Bernoulli($\theta$) random variables. Find the distribution of $Y = \sum_{i=1}^n X_i$. 

\item Let $X_1, \ldots, X_n$ be independent Normal($\mu,\sigma^2$) random variables. Find the distribution of the sample mean $\overline{X} = \frac{1}{n}\sum_{i=1}^n X_i$. 

\item Let $X_1, \ldots, X_n$ be independent Gamma($\alpha,\lambda$) random variables. Find the distribution of the sample mean $\overline{X} = \frac{1}{n}\sum_{i=1}^n X_i$. 

\item Let $Z \sim N(0,1)$ and let $Y = Z^2$. Find the distribution of  $Y$ using moment-generating functions.

\item \label{jlast} Let $X$ be a \emph{degenerate} random variable with $P(X=\mu) = 1$.
    \begin{enumerate}
        \item Find the moment-generating function.
        \item Differentiate to obtain $E(X)$ and $Var(X)$. Do these answers make sense?
        \item Comparing this to the moment-generating function of a normal, one can say that in a weird way, a degenerate distribution is normal with variance \underline{\hspace{10mm}}.
    \end{enumerate}

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\item \label{L1} Suppose that $X$ and $Y$ are discrete independent random variables with the following moment generating functions:
$$M_X(t)=E(e^{tX})=\frac{1}{6}e^{t}+\frac{2}{6}e^{2t}+\frac{3}{6}e^{3t}$$
$$M_Y(t)=E(e^{tY})=\frac{1}{10}e^{-t}+\frac{4}{10}e^{2t}+\frac{1}{2}e^{3t}.$$
Using the moment generating function, find the distribution of 
    \begin{enumerate}
        \item $Z=X+Y$.
        \item  $U=X-Y$.
    \end{enumerate}

\item  Let $X$ and $Y$ be discrete random variables such that $$p_X(x)=\frac{1}{3}, \ \ x=-1,0,1$$ and $$p_Y(y)=\frac{1}{2}, \ \ y=2,4.$$
Let $Z=X+Y$.
\begin{enumerate}
    \item  Using the probability mass functions of $X$ and $Y$, find the probability mass function of $Z$.
    \item  Find the moment generating function of $Z$.
    \item  Using part (b), find the probability mass function of $Z$. Does your answer agree with (a)?
\end{enumerate}

\item  Let $X$ and $Y$ be independent random variables, both with $Poisson(\lambda)$ distribution, for some $\lambda>0$. Define $Z=X+Y$.
\begin{enumerate}
    \item Find  the distribution of $Z$ by using the moment generating function.
    \item For any non-negative integer $n$, find the conditional probability mass function  of $X$ given $Z=n$.
    \item State the name of the conditional distribution of $X$ given $Z=n$.
\end{enumerate}
    
\item  Let $X$ be a continuous random variable with pdf $f(x)=k e^{-|x|}, -\infty<x<\infty$.

\begin{enumerate}
    \item Find the value of the constant $k$.
    \item Find the moment generating function of $X$.
    \item Find the mean and the variance of $X$.
\end{enumerate}

\item Let $M_{X}(t)$ be the moment generating function of a random variable $X$.
\begin{enumerate}
    \item    Show that $M_{X}(t)=E(e^{tX})=\sum_{k=0}^\infty E(X^k) \frac{t^k}{k!}$. \textbf{Hint:} $e^x=\sum_{k=0}^\infty \frac{x^k}{k!}$.
    \item   It follows form part (a) that, in Maclaurin (power) series, $E(X^k)$ is the coefficient of $t^k/k!$, $k=0,1,2,\ldots$. Use this fact to find $E(X^{1024})$ in the following  cases:
        \begin{enumerate}
            \item $M_{X}(t)=\frac{1}{1-t^2}$, $|t|<1$.
            \item $M_{X}(t)=e^{t^2}$.
            \item $M_{X}(t)=\frac{1}{(1-5t)^2}$, $|5t|<1$.
        \end{enumerate}
    \end{enumerate}

\item  Let $M_{X}(t)$ be the moment generating function of a random variable $X$. Define $S(t)=\log M_X(t)$.

\begin{enumerate}
    \item Find $S(0)$
    \item Show that $\frac{d}{dt}S(t)|_{t=0}=E(X)$. 
    \item Show that $\frac{d^2}{dt^2}S(t)|_{t=0}=Var(X)$. 
\end{enumerate}
    
\item Let $X \sim N(\mu=1, \sigma^2=4)$. If $Y=0.5^X$, find $E(Y^2)$. \textbf{Hint:} Use moment generating function.

\item \label{Llast} If $M_X(t)=e^{3t+8t^2}$ is the moment generating function of the random variable $X$, find $P(-1<X<9)$.


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\noindent
Questions \ref{j1} through \ref{jlast} were prepared by  \href{http://www.utstat.toronto.edu/~brunner}{Jerry Brunner},
Department of Mathematical and Computational Sciences, University of Toronto. They are licensed under a 
\href{http://creativecommons.org/licenses/by-sa/3.0/deed.en_US}
     {Creative Commons Attribution - ShareAlike 3.0 Unported License}. Questions \ref{L1} through \ref{Llast} were prepared by Luai Al Labadi, Department of Mathematical and Computational Sciences, University of Toronto. I am not sure what his preferences are, so all rights to Luia's questions are reserved. The \LaTeX~source code is available from the course website: 
     
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