\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?
\nosolspace{1.5cm}
\end{Parts}
\Question{More Family Planning}
\begin{Parts}
\Part Suppose we have a random variable $N \sim Geom(1/3)$ representing the number of children of a randomly chosen family. Assume that within the family, children are equally likely to be boys and girls. Let $B$ be the number of boys and $G$ the number of girls in the family. What is the joint probability distribution of $B$, $G$?
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\Part Given that we know there are $0$ girls in the family, what is the most likely number of boys in the family?
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\Part Now let $X$ and $Y$ be independent random variables representing the number of children in two independently, randomly chosen families. Suppose that $X \sim Geom(p)$ and $Y \sim Geom(q)$. Find $\Pr(X < Y)$, the probability that the number of children in the first family ($X$) is less than the number of children in the second family ($Y$).
(You may use the convergence formula for a Geometric Series: $\sum_{k=0}^\infty r^k = \frac{1}{1-r}$ for $|r| < 1$)
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\Part Show how you could obtain your answer from the previous part using an interpretation of the geometric distribution.
\nosolspace{1.5cm}
\end{Parts}
\Question {Combining Distributions}
\begin{Parts}
%\Part
%Define $X \sim Bin(n, p)$ and $Y \sim Bin(m, p)$, where $X$ and $Y$ are independent. Prove that $X + Y$ also has a binomial distribution. \\
%(\textit{Hint}: Use a combinatorial proof to simplify some expressions.)
\Part Let $X \sim Pois(\lambda), Y \sim Pois(\mu)$ be independent. Prove that $X + Y \sim Pois(\lambda + \mu)$. \\
\textit{Hint}: Recall the binomial theorem, which states that
\[
(a + b)^n = \sum_{i = 0}^n \binom{n}{i}a^ib^{n - i}.
\]
\Part
Let $X$ and $Y$ be defined as in the previous part. Prove that the distribution of $X$ conditional on $X+Y$ is a binomial distribution, e.g. that $X|X+Y$ is binomial. What are the parameters of the binomial distribution? \\
\textit{Hint}: Your result from the previous part will be helpful.
\end{Parts}
\Question{Darts}
Yiming is playing darts.
Her aim follows an exponential distribution with parameter 1; 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 Yiming made it on the board. What is the probability she is within 1 unit from the center?
\Part If Yiming is within 1 unit from the center, she scores 4 points, if she is within 2 units, she scores 3, etc. In other words, Yiming scores $\lfloor 5 - x\rfloor$, where $x$ is the distance from the center. (This implies that Yimin scores 0 points if she throws it off the board). What is Yiming's expected score after one throw?
\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.]
\nosolspace{2cm}
\end{Parts}
\Question{Moments of the Exponential Distribution}
Let $X \sim \Expo(\lambda)$, where $\lambda > 0$.
Show that for all positive integers $k$, $\E[X^k] = k!/\lambda^k$.
[\textit{Hint: }Integration by Parts.]