\Question{Proof with Indicators}
Let $n \in \Z_+$.
Let $\alpha_1, \dotsc, \alpha_n \in \R$ and let $A_1, \dotsc, A_n$ be events.
Prove that $\sum_{i=1}^n \sum_{j=1}^n \alpha_i \alpha_j \Pr(A_i \cap A_j) \ge 0$.
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\Question{Balls and Bins}
Throw $n$ balls into $m$ bins, where $m$ and $n$ are positive integers. Let $X$ be the number of bins with exactly one ball. Compute $\var X$.
\Question{Portfolio Optimization}
Suppose that there are $n$ assets, where $n$ is a positive integer.
For each unit dollar invested in asset $i$, for $i=1,\dotsc,n$, with probability $p_i$ the value of the asset will grow by $\alpha_i$ to $1+\alpha_i$, and with probability $1-p_i$ the value of the asset will shrink by $\alpha_i$ to $1 - \alpha_i$.
Let the proportion of money invested in asset $i$ be $w_i$ (so that $\sum_{i=1}^n w_i = 1$), and let $X_i$ be a random variable denoting the final value of the $i$th asset per unit dollar. Then $X = w_1X_1 + \cdots + w_nX_n$ is the total value. For simplicity, assume that the outcomes of the different assets are independent.
\begin{Parts}
\Part Compute the expectation $\E[X]$. What values of $w_i$ maximize this quantity?
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\Part Compute the variance $\var X$. What values of $w_i$ minimize this quantity?
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\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 $Y \geq 0$.]
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\Part Let $Z = \max\{X_1, X_2, \dotsc, X_n\}$. Find $\E(Z)$.
[\textit{Hint}: Find the CDF.]
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\end{Parts}
\Question{Darts (Again!)}
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 Distributions: Lightbulbs}
A brand new lightbulb has just been installed in our classroom, and
you know the life span of a lightbulb is exponentially distributed
with a mean of $50$ days.
\begin{enumerate}[(a)]
\item Suppose an electrician is scheduled to check on the lightbulb in $30$
days and replace it if it is broken. What is the probability that
the electrician will find the bulb broken?
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\item Suppose the electrician finds the bulb broken and replaces it with
a new one. What is the probablity that the new bulb will last at least
$30$ days?
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\item Suppose the electrician finds the bulb in working condition and leaves.
What is the probability that the bulb will last at least another $30$
days?
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\end{enumerate}