\Question{Binomial Variance}
Throw $n$ balls into $m$ bins uniformly at random. For a specific ball $i$, what is the variance of the number of roommates it has (i.e.\ the number of other balls that it shares its bin with)?
\Question{Working with Distributions}
\begin{enumerate}
\item
For each of the following scenarios, describe the sample space and find the associated probabilities for each random variable.
\begin{enumerate}
\item Five fair coins are flipped and the random variable $Y$ is defined as the number of tails observed.
\item Two dice are rolled and the random variable $Z$ is defined as the product of the two numbers rolled.
\end{enumerate}
\item
Suppose a fair six sided dice is rolled until a number smaller than 3 is observed. Let $N$ be the total number of times the dice is rolled. Find $\Pr(N = k)$ for $k = 1, 2, 3, \dotsc$.
\item
Now suppose two six-sided dice are rolled and the two numbers observed are defined as $X$ and $Y$.
\begin{enumerate}
\item Calculate $\Pr(X > 3 \mid Y = 1)$.
\item Let $Z = X+Y$. What is the range of $Z$?
\item Calculate $\Pr(X = 1 \mid Z < 4)$.
\end{enumerate}
\end{enumerate}
\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{Exploring the Geometric Distribution}
\begin{Parts}
\Part Let $X \sim \operatorname{Geometric}(p)$ and $Y \sim \operatorname{Geometric}(q)$ are independent. Find the distribution of
$\min\{X, Y\}$ and justify your answer.
\Part Let $X$, $Y$ be i.i.d.\ geometric random variables with parameter $p$.
Let $U = \min\{X, Y\}$ and $V = \max\{X, Y\} - \min\{X, Y\}$.
Compute the joint distribution of $(U, V)$
\Part Prove that $U$ and $V$ are independent.
\end{Parts}
\Question{Boutique Store}
Consider a boutique store in a busy shopping mall. Every hour, a large number of people visit the mall, and each independently enters the boutique store with some small probability. The store owner decides to model $X$, the number of customers that enter her store during a particular hour, as a Poisson random variable with mean $\lambda$.
Suppose that whenever a customer enters the boutique store, they leave the shop without buying anything with probability $p$. Assume that customers act independently, i.e.~you can assume that they each flip a biased coin to decide whether to buy anything at all. Let us denote the number of customers that buy something as $Y$ and the number of them that do not buy anything as $Z$ (so $X = Y+Z$).
\begin{Parts}
\Part What is the probability that $Y=k$ for a given $k$? How about $\Pr[Z=k]$?
\textit{Hint}: You can use the identity
\begin{align*}
\e^x=\sum_{k=0}^{\infty}\frac{x^k}{k!}.
\end{align*}
\Part State the name and parameters of the distribution of $Y$ and $Z$.
\Part Prove that $Y$ and $Z$ are independent.
\end{Parts}
\Question{Student Life}
In an attempt to avoid having to do laundry often, Marcus comes up with a system. Every night, he designates one of his shirts as his dirtiest shirt. In the morning, he randomly picks one of his shirts to wear. If he picked the dirtiest one, he puts it in a dirty pile at the end of the day (a shirt in the dirty pile is not used again until it is cleaned). When Marcus puts his last shirt into the dirty pile, he finally does his laundry, and again designates one of his shirts as his dirtiest shirt (laundry isn't perfect) before going to bed. This process then repeats.
\begin{Parts}
\Part If Marcus has $n$ shirts, what is the expected number of days that transpire between laundry events? Your answer should be a function of $n$ involving no summations.
\Part Say he gets even lazier, and instead of organizing his shirts in his dresser every night, he throws his shirts randomly onto one of $n$ different locations in his room (one shirt per location), designates one of his shirts as his dirtiest shirt, and one location as the dirtiest location. In the morning, if he happens to pick the dirtiest shirt, \textit{and} the dirtiest shirt was in the dirtiest location, then he puts the shirt into the dirty pile at the end of the day and does not use that location anymore (it is too dirty now). What is the expected number of days that transpire between laundry events now? Again, your answer should be a function of $n$ involving no summations.
\end{Parts}