\Question{Safeway Monopoly Cards}
It's that time of the year again - Safeway is offering its Monopoly Card promotion. Each time you visit Safeway, you are given one of $n$ different Monopoly Cards with equal probability. You need to collect them all to redeem the grand prize.
%\begin{enumerate}[(a)]
Let $X$ be the number of visits you have to make before you can redeem the grand prize. Show that $\Var{X} = n^2\left(\sum_{i=1}^n i^{-2}\right) - \E(X)$. \textit{[Hint: Does this remind you of a particular problem? What is the expectation for this problem?]}
% \item The series $\sum_{i=1}^\infty i^{-2}$ converges to the constant value $\pi^2/6$. Using this fact and Chebyshev's Inequality, find a lower bound on $\beta$ for which the probability you need to make more than $\E(X) + \beta n$ visits is less than $1/100$, for large $n$. \textit{[Hint: Use the approximation $\sum_{i = 1}^n i^{-1} \approx \ln n$ as $n$ grows large.]}
%\end{enumerate}
\Question{Geometric Distribution}
Two faulty machines, $M_1$ and $M_2$, are repeatedly run synchronously in parallel (i.e., both machines execute one run, then both execute a second run, and so on). On each run, $M_1$ fails with probability $p_1$ and $M_2$ fails with probability $p_2$, all failure events being independent. Let the random variables $X_1$, $X_2$ denote the number of runs until the first failure of $M_1$, $M_2$ respectively; thus $X_1$, $X_2$ have geometric distributions with parameters $p_1$, $p_2$ respectively. Let $X$ denote the number of runs until the first failure of \emph{either} machine.
\begin{Parts}
\Part Show that $X$ also has a geometric distribution, with parameter $p_1+p_2-p_1p_2$.
\Part Now, two technicians are hired to check on the machines every run. They decide to take turns checking on the machines every hour. What is the probability that the first technician is the first one to find a faulty machine?
\end{Parts}
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\Question{Geometric and Poisson}
Let $X$ be geometric with parameter $p$, $Y$ be Poisson with parameter $\lambda$, and $Z=\max(X,Y)$. Assume $X$ and $Y$ are independent. For each of the following parts, your final answers should not have summations.
\begin{Parts}
\Part Compute $P(X>Y)$.
\Part Compute $P(Z\geq X)$.
\Part Compute $P(Z\leq Y)$.
\end{Parts}
\Question{Darts}
Alvin is playing darts.
His aim follows an exponential distribution; that is, the probability density that the dart is $x$ distance from the center is $f_X(x) = \exp(-x)$.
The board's radius is 4 units.
\begin{Parts}
\Part What is the probability the dart will stay within the board?
\Part Say you know Alvin made it on the board. What is the probability he is within 1 unit from the center?
\Part If Alvin is within 1 unit from the center, he scores 4 points, if he is within 2 units, he scores 3, etc. In other words, Alvin scores $\lfloor 5 - x\rfloor$, where $x$ is the distance from the center. What is Alvin's expected score after one throw?
\end{Parts}
\Question{Exponential Practice}
\begin{Parts}
\Part Let $X_1, X_2 \sim \operatorname{Exponential}(\lambda)$ be independent, $\lambda > 0$.
Calculate the density of $Y := X_1 + X_2$.
[\textit{Hint}: One way to approach this problem would be to compute the CDF of $Y$ and then differentiate the CDF.]
\Part Let $t > 0$.
What is the density of $X_1$, conditioned on $X_1 + X_2 = t$?
[\textit{Hint}: Once again, it may be helpful to consider the CDF $\Pr(X_1 \le x \mid X_1 + X_2 = t)$.
To tackle the conditioning part, try conditioning instead on the event $\{X_1 + X_2 \in [t, t + \varepsilon]\}$, where $\varepsilon > 0$ is small.]
\end{Parts}
\Question{Uniform Means}
Let $X_1, X_2, \dotsc, X_n$ be $n$ independent and identically distributed uniform random variables on the interval $[0, 1]$ (where $n$ is a positive integer).
\begin{Parts}
\Part Let $Y = \min\{X_1, X_2, \dotsc, X_n\}$. Find $\E(Y)$. [\textit{Hint}: Use the tail sum formula, which says the expected value of a nonnegative random variable is $\E(X) = \int_0^{\infty} \Pr(X > x) \, \D x$. Note that we can use the tail sum formula since $Y \geq 0$.]
\nosolspace{2cm}
\Part Let $Z = \max\{X_1, X_2, \dotsc, X_n\}$. Find $\E(Z)$.
[\textit{Hint}: Find the CDF.]
\nosolspace{2cm}
\end{Parts}