# Part 1: Probability distributions

\begin{enumerate}
\item A contestant on a game show needs to answer 10 questions correctly to win the jackpot. However, if they get 4 incorrect answers, they are kicked off the show. Suppose one contestant consistently has a 80\% chance of correctly responding to any question. 

\begin{enumerate}
\item What is the probability distribution?
\item What is the probability that she will correctly answer 10 questions before 4 incorrect responses? 
\item Write out the R code to calculate (b).
\end{enumerate}


\item A small town’s police department issues 5 speeding tickets per month on average. 

\begin{enumerate}
\item Using a Poisson random variable, what is the likelihood that the police department issues 3 or fewer tickets in one month? 
\item What is the probability that 10 days or fewer elapse between two tickets being issued?
\item Write out the R code to calculate (a), (b).
\end{enumerate}

\end{enumerate}

# Part 2: Statistical inference

\begin{enumerate}
\item (AoS 6.6.2) Let $X_{1}, \ldots, X_{n} \sim \operatorname{Uniform}(0, \theta)$ and let $\hat{\theta}=\max \left\{X_{1}, \ldots, X_{n}\right\}$. Find the bias, se and MSE of this estimator.

\item (AoS 6.6.3) Let $X_{1}, \ldots, X_{n} \sim \operatorname{Uniform}(0, \theta)$ and let $\hat{\theta}=2 \bar{X}_{n}$. Find the bias, se and MSE of this estimator.

\item Let $X_{1}, \ldots, X_{n} \sim \operatorname{Uniform}(0,1)$. Let $Y_{n}=\bar{X}_{n}^{2}$. Find the limiting distribution of $Y_{n}$. (Hint: CLT)

\end{enumerate}


