\Question{Waiting For the Bus}
Edward and Jerry are waiting at the bus stop outside of Soda Hall.
Like many bus systems, buses arrive in periodic intervals. However, the Berkeley
bus system is unreliable, so the length of these intervals are random, and follow Exponential
distributions.
Edward is waiting for the 51B, which arrives according to an Exponential distribution
with parameter $\lambda$. That is, if we let the random variable $X_i$ correspond
to the difference between the arrival time $i$th and $i-1$st bus (also known as the inter-arrival time)
of the 51B, $X_i \sim \operatorname{Expo}(\lambda)$.
Jerry is waiting for the 79, whose inter-arrival time, follows an Exponential distributions
with parameter $\mu$. That is, $Y_i \sim \operatorname{Expo}(\mu)$. Assume that all inter-arrival times are independent.
\begin{Parts}
\Part What is the probability that Jerry's bus arrives before Edward's bus?
\Part After 20 minutes, the 79 arrives, and Jerry rides the bus. However, the 51B still hasn't arrived yet. Let
Let $D$ be the additional amount of time Edward needs to wait for the 51B to arrive. What is the distribution of $D$?
\Part Lavanya isn't picky, so she will wait until either the 51B or the 79
bus arrives. Solve for the distribution of $Z$, the amount of time Lavanya
will wait before catching the bus.
\Part Khalil arrives at the bus stop, but he doesn't feel like riding the bus with Edward. He
decides that he will wait for the second arrival of the 51B to ride the bus. Find the distribution
of $T = X_1 + X_2$, the amount of time that Khalil will wait to ride the bus.
[\textit{Hint}: One way to approach this problem would be to compute the CDF of $T$ and then differentiate the CDF.]
\end{Parts}
\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{Normal Darts?}
Alex and John are playing a game of darts. Let $(X_a, Y_a)$ and $(X_j, Y_j)$ denote the coordinates of Alex's and John's darts on the board and are distributed in the following way:
\begin{enumerate}
\item $X_a, Y_a \sim \mathbb{N} (0, 1)$ independently
\item $X_j, Y_j$ are distributed uniformly in a circle of radius $3$
\end{enumerate}
The winner of the game is determined by whoever's darts is closer to the center of the board at $(0,0)$. In this question, we will compute the probability that Alex wins the game. We will denote the squared distances of the darts from the center by $r_a = X_a^2 + Y_a^2$ and by $r_j = X_j^2 + Y_j^2$.
\begin{Parts}
\Part What is the distribution of $r_a$? \\
Hint: Consider the joint distribution and the following change of variables formula: Suppose we want to integrate the function $f(x,y)$ over the circle $\left(\sqrt{x^2 + y^2} \leq R\right)$. Then, we have the following change of variables formula:
\begin{equation*}
\int_{\sqrt{x^2 + y^2} \leq R} f(x,y) dx dy = \int_{0}^{R} \int_{0}^{2\pi} f(r\cos \theta, r \sin \theta) r d\theta dr
\end{equation*}
You may find the identity $\sin(\theta)^2 + \cos(\theta)^2 =1$ useful.
\Part What is the distribution of $r_j$? (Hint: Try computing the CDF first)
\Part What is the probability that Alex wins the game?
\end{Parts}
% written Spr 16
\Question{The Dwinelle Labyrinth}
You have decided to take a humanities class this semester, a French class to be specific. Instead of a final exam, your professor has issued a final paper. You must turn in this paper \textit{before} noon to the professor's office on floor 3 in Dwinelle, and it's currently 11:48 a.m.
Let Dwinelle be modeled by the following Markov chain. Instead of rushing to turn it in, we will spend valuable time computing whether or not we \textit{could have} made it. Suppose walking between floors takes 1 minute.
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=2.8cm,
semithick]
\tikzstyle{every state}=[fill=white,draw=black,thick,text=black,scale=1]
\node[state] (A) {$F_1$};
\node[state] (B) [right of=A] {$F_2$};
\node[state] (C) [below of=B] {$F_3$};
\node[state] (D) [below of=A] {$F_0$};
\node[state] (E) [right of=B] {$F_4$};
\path (B) edge [bend right] (A);
\path (B) edge [bend left] (C);
\path (A) edge [bend left] (D);
\path (B) edge [bend right] (D);
\path (D) edge [bend right] (B);
\path (D) edge [bend left] (A);
\path (B) edge [bend right] (E);
\path (E) edge [bend right] (B);
\end{tikzpicture}
\begin{Parts}
\Part Will you make it in time if you choose a floor to transition to uniformly at random?
(If $T_i$ is the number of steps needed to get to $F_3$ starting from $F_i$, where $i \in \{0, 1, 2, 3, 4\}$, is $\E[T_0]$ < 12?)
\nosolspace{5cm}
\Part Will you make it in time, if for every floor, you order all accessible floors and are twice as likely to take higher floors? (If you are considering $1$, $2$, or $3$, you will take each with probabilities $1/7$, $2/7$, $4/7$, respectively.)
\nosolspace{5cm}
\end{Parts}
\Question{Faulty Machines}
You are trying to use a machine that only works on some days. If on a given day the machine is working, it will break down the next day with probability $0 < b < 1$, and works on the next day $1 - b$. If it is not working on a given day, it will work on the next day with probability $0 < r < 1$, and not work on the next day with probability $1 - r$. Formulate this process as a Markov chain. As $n \rightarrow \infty$, what does the probability that on a given day the machine is working converge to? What properties of the Markov chain allow us to conclude that the probability will actually converge?
\Question{Three Tails}
You flip a fair coin until you see three tails in a row. What is the average number of heads that you'll see until getting $TTT$?