\Question{Family Planning}
Mr. and Mrs. Brown decide to continue having children until they either have their first girl or until
they have three children. Assume that each child is equally likely to be a boy or a girl, independent of
all other children, and that there are no multiple births. Let $G$ denote the numbers of girls that the Browns have. Let $C$ be the total number of children they have.
\begin{Parts}
\Part Determine the sample space, along with the probability of each sample point.
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\Part Compute the joint distribution of $G$ and $C$. Fill in the table below.
\scalebox{1.2}{
\begin{tabular}{|c||c|c|c|}
\hline
& $C = 1$ & $C = 2$ & $C = 3$ \\
\hline
\hline
$G = 0$ & & & \\
\hline
$G = 1$ & & & \\
\hline
\end{tabular}}
\Part Use the joint distribution to compute the marginal distributions of $G$ and $C$ and confirm that the values are as you'd expect. Fill in the tables below.
\scalebox{1.2}{
\begin{tabular}{|c||m{.85cm}|}
\hline
$\Pr(G = 0)$ & \\
\hline
$\Pr(G = 1)$ & \\
\hline
\end{tabular}}
\scalebox{1.2}{
\begin{tabular}{|c|c|c|}
\hline
$\Pr(C = 1)$ & $\Pr(C = 2)$ & $\Pr(C = 3)$ \\
\hline
\hline
& & \\
\hline
\end{tabular}}
\Part Are $G$ and $C$ independent?
\Part What is the expected number of girls the Browns will have? What is the expected number of children that the Browns will have?
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\end{Parts}
\Question{Will I Get My Package?}
A delivery guy in some company is out delivering $n$ packages to $n$ customers, where $n \in \N$, $n > 1$.
Not only does he hand a random package to each customer, he opens the package before delivering it with probability $1/2$.
Let $X$ be the number of customers who receive their own packages unopened.
\begin{Parts}
\Part Compute the expectation $\E(X)$.
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\Part Compute the variance $\var(X)$.
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\end{Parts}
\Question{Double-Check Your Intuition Again}
\begin{Parts}
\Part You roll a fair six-sided die and record the result $X$. You roll the die again and record the result $Y$.
\begin{enumerate}[(i)]
\item What is $\cov (X+Y, X-Y)$?
\item Prove that $X+Y$ and $X-Y$ are not independent.
\end{enumerate}
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For each of the problems below, if you think the answer is "yes" then provide a proof. If you think the answer is "no", then provide a counterexample.
\Part If $X$ is a random variable and $\var (X) = 0$, then must $X$ be a constant?
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\Part If $X$ is a random variable and $c$ is a constant, then is $\var (cX) = c \var (X)$?
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\Part If $A$ and $B$ are random variables with nonzero standard deviations and $\text{Corr} (A, B) = 0$, then are $A$ and $B$ independent?
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\Part If $X$ and $Y$ are not necessarily independent random variables, but $\text{Corr} (X, Y) = 0$, and $X$ and $Y$ have nonzero standard deviations, then is $\var (X+Y) = \var(X) + \var(Y)$?
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\Part If $X$ and $Y$ are random variables then is $\E(\max (X, Y) \min (X, Y)) = \E(X Y)$?
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\Part If $X$ and $Y$ are independent random variables with nonzero standard deviations, then is $$\text{Corr} (\max (X, Y), \min (X, Y)) = \text{Corr} (X, Y) ?$$
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\end{Parts}