\Question{Darts with Friends}
Michelle and Alex are playing darts.
Being the better player, Michelle's aim follows a uniform distribution over a circle of radius $r$ around the center. Alex's aim follows a uniform distribution over a circle of radius $2r$ around the center.
\begin{Parts}
\Part Let the distance of Michelle's throw be denoted by the random variable $X$ and let the distance of Alex's throw be denoted by the random variable $Y$.
\begin{itemize}
\item What's the cumulative distribution function of $X$?
\item What's the cumulative distribution function of $Y$?
\item What's the probability density function of $X$?
\item What's the probability density function of $Y$?
\end{itemize}
\Part What's the probability that Michelle's throw is closer to the center than Alex's throw? What's the probability that Alex's throw is closer to the center?
\Part What's the cumulative distribution function of $U = \min\{X,Y\}$?
\Part What's the cumulative distribution function of $V = \max\{X,Y\}$?
\Part What is the expectation of the absolute difference between Michelle's and Alex's distances from the center, that is, what is $\E[|X - Y|]$?
[\textit{Hint}: There are two ways of solving this part.]
\end{Parts}
\Question{Variance of the Minimum of Uniform Random Variables}
Let $n$ be a positive integer and let $X_1,\dotsc,X_n \overset{\text{i.i.d.}}{\sim} \Unif[0, 1]$.
Find $\var Y$, where
$$Y := \min\{X_1,\dotsc,X_n\}.$$
\Question{Exponential Practice}
Let $X \sim \operatorname{Exponential}(\lambda_X)$ and $Y \sim \operatorname{Exponential}(\lambda_Y)$ be independent, where $\lambda_X, \lambda_Y > 0$.
%Let $\one\{X \le Y\}$ and $\one\{Y \le X\}$ denote the indicators for the events $\{X \le Y\}$ and $\{Y \le X\}$ respectively.
Let $U = \min\{X, Y\}$, $V = \max\{X, Y\}$, and $W = V - U$.
\begin{Parts}
\Part Compute $\Pr(U > t, X \le Y)$, for $t \ge 0$.
\Part Use the previous part to compute $\Pr(X \le Y)$.
Conclude that the events $\{U > t\}$ and $\{X \le Y\}$ are independent.
\Part Compute $\Pr(W > t \mid X \le Y)$.
\Part Use the previous part to compute $\Pr(W > t)$.
\Part Calculate $\Pr(U > u, W > w)$, for $w > u > 0$.
Conclude that $U$ and $W$ are independent.
[\textit{Hint}: Think about the approach you used for the previous parts.]
\end{Parts}
\Question{Exponential Practice II}
\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{Moments of the Exponential Distribution}
Let $X \sim \Expo(\lambda)$, where $\lambda > 0$.
Show that for all positive integers $k$, $\E[X^k] = k!/\lambda^k$.
[Use induction.]
\Question{Exponential Approximation to the Geometric Distribution}
Say you want to buy your friend a gift for her birthday but it totally slipped your mind and your friend's birthday already passed.
Well, better late than never, so you order a package from Amazon which will arrive in $X$ seconds, where $X \sim \Geom(p)$ (for $p \in (0, 1)$).
The later the package arrives, the worse it is, so say that the cost of giving your friend the gift is $X^4$, and you wish to compute $\E[X^4]$.
Unfortunately, computing $\E[X^4]$ is very tedious for the geometric distribution, so approximate $X$ by a suitable exponential distribution and compute $\E[X^4]$.
To get started on this problem, take a look at $\Pr(X > x)$ for the geometric distribution and the exponential distribution and note the similarity.
[\textit{Note}: This problem illustrates that sometimes, moving to the continuous world simplifies calculations!]