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\begin{center}   {\Large \textbf{STA 256f18 Assignment Ten}}\footnote{Copyright information is at the end of the last page.}\vspace{1 mm}\end{center}\noindent
Please read Chapter 5 in the text.
The following homework problems are not to be handed in. They are preparation for  the final exam.  All textbook problems are from Chapter Five. 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 or derive it.
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\begin{enumerate} 

%%%%%%%%%%%%%%%%%%%% Convergence in probability and LLN %%%%%%%%%%%%%%%%%%%%
% Including some conditional expected value in the first 2 questions. 

\item \label{beta} For $n = 1, 2, \ldots$, let $X_n$ have a beta distribution with $\alpha=n$ and $\beta=1$. % , so that the density of $X_n$ is $nx^{n-1}$ for $0\leq x \leq 1$, and zero otherwise. 
    \begin{enumerate}
        \item Find the cumulative distribution function of $X_n$.
        \item Show that $X_n$ converges in probability to a constant, and find the constant. Convergence in probability means for all $\epsilon > 0$,  $\lim_{n \rightarrow \infty}P\{|X_n-c|\geq\epsilon\} = 0 $.
    \end{enumerate}
    
\item Use Chebyshev's inequality to prove the Law of Large Numbers.

\item The continuous mapping theorem for convergence in probability says that if $g(x)$ is a function that is continuous at $x=c$, and if $T_n$ converges in probability to $c$, then $g(T_n)$  converges in probability to $g(c)$. A gamma random variable has expected value $\alpha/\lambda$, something you could easily show if you needed to. Let $X_1, \ldots, X_n$ be a collection of independent gamma random variables with unknown parameter $\alpha$, and known $\lambda=6$. Find a random variable $T_n=g(\overline{X}_n)$ that converges in probability to $\alpha$. The statistic $T_n$ is a good way to estimate $\alpha$ from sample data.


%%%%%%%%%%%%%%%%%%%% Convergence in distribution and CLT %%%%%%%%%%%%%%%%%%%%

% A mixture of those 2 betas would be nice. -> Bernoulli

\item Consider a \emph{degenerate} random variable $X$, with $P(X=c)=1$. 
    \begin{enumerate}
        \item What is $F_x(x)$, the cumulative distribution function of $X$? Your answer must apply to all real $x$.
        \item Give a formula for $M_x(t)$, the moment-generating function of $X$. 
    \end{enumerate}

\item Recall that convergence of $X_n$ to $X$ in distribution means that $F_n(x) \rightarrow F(x)$ as $n \rightarrow \infty$ for all continuity points of $F(x)$. For the distribution of Question~\ref{beta}, 
    \begin{enumerate}
        \item What is $\lim_{n \rightarrow \infty} F_{x_n}(x)$ for $x<1$?
        \item What is $\lim_{n \rightarrow \infty} F_{x_n}(x)$ for $x>1$?
        \item What do you conclude?
    \end{enumerate}

\item Sometimes, a sequence of random variables does not converge in distribution to anything. Let $X_n$ have a continuous uniform distribution on $(0,n)$. Clearly, $\lim_{n \rightarrow \infty} F_{x_n}(x)=0$ for $x \leq 0$. Find $\lim_{n \rightarrow \infty} F_{x_n}(x)$ for $x>0$. Is $\lim_{n \rightarrow \infty} F_{x_n}(x)$ continuous? Is it a cumulative distribution function? 

\item Let $X_n$ be a binomial ($n,p_n$) random variable with $p_n=\frac{\lambda}{n}$, so that $n \rightarrow \infty$ and $p \rightarrow 0$ in such a way that the value of $n \, p_n=\lambda$ remains fixed. Using moment-generating functions, find the limiting distribution of $X_n$.

\item Let $X_1, \ldots, X_n$ be independent geometric random variables, so that $E(X_i) = \frac{1}{p}$ and $Var(X_i)=\frac{1-p}{p^2}$. If $p=1/2$ and $n=64$, find the approximate $P(\overline{X}_n) > 2.5$. Answer: 0.0023

\item Let $X_1, \ldots, X_n$ be independent random variables from an unknown distribution with expected value 5.1 and standard deviation 4.8. Find the approximate probability that the sample mean will be greater than 6 for $n=25$. Answer: 0.1736

\item The ``normal approximation to the binomial" says that if $X \sim$ Binomial($n,p$), then for large $n$,
\begin{displaymath}
    Z = \frac{X-np}{\sqrt{np(1-p)}}
\end{displaymath}
may be treated as standard normal to obtain approximate probabilities. Where does this formula come from? Hint: What is the distribution of a sum of independent Bernoulli random variables?



\end{enumerate} % End of questions
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\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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