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\title{\sc Math 20 -- Homework 7}
\author{due Wednesday, August 16}
\date{}
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\textbf{Instructions:} This assignment is due at the \emph{beginning} of class. Staple your work together (do not just fold over the corner). Please write the questions in the correct order. If I cannot read your handwriting, you won't receive full credit.\\
%\emph{You may use Wolfram Alpha to compute any necessary sums or integrals. If you have trouble with this, let me know.}\\
\textbf{If you're using facts about distributions to answer the questions, be very clear about which distribution you're using to model that problem and why that distribution is appropriate.}
\begin{enumerate}
\item Suppose that the height, in inches, of a 25-year old man is a normal random variable with parameters $\mu = 71$ and $\sigma^2 = 6.25$. What percentage of $25$-year old men are over $6$ feet $2$ inches tall? What percentage of men over $6$ feet tall are over $6$ foot $5$ inches?
\item Suppose that $X$ and $Y$ are two independent random variables with density functions
\[
f_X(x) = \left\{\begin{array}{ll}
\frac{3}{2}(x+1)^2,&\text{ if }-1 \leq x \leq 0\\[5pt]
\frac{3}{2}(x-1)^2,&\text{ if }0 \leq x \leq 1\\[5pt]
0,&\text{otherwise}
\end{array}\right.
\]
and
\[
f_Y(x) = \left\{\begin{array}{ll}
\frac{1}{2},&\text{ if } 0 \leq x \leq 2\\[5pt]
0,&\text{otherwise}
\end{array}\right..
\]
Find the density function for $Z = X + Y$.
\item In this exercise, you will use the notion of convolution and the principle of mathematical induction (a proof technique) to prove that if $X_1, \ldots, X_n$ are independent and identically distributed exponential random variables with rate $\lambda$, then the probability density function of $S_n = X_1 + \cdots + X_n$ (for $n \geq 1$) is
\[
f_{S_n}(x) = \frac{\lambda e^{-\lambda x} (\lambda x)^{n-1}}{(n-1)!}.
\]
Note that we proved the $n=2$ case explicitly in class.
\textbf{Mathematical Induction:} Suppose you have a statement that you want to prove is true for all positive integers $n = 1,2,3,\ldots$. One way to do this is induction: first you show it's true for $n=1$ (this is called the base case), then you show that \emph{if} it's true for some number $k$ \emph{then} it must be true for the next number $k+1$ (this is called the induction step). If you show these two things, then the statement must be true for all $n$. (Why? Because you showed it's true for $1$ in the base case. Then the induction step shows that if it's true for $1$ then it must be true for $2$. If it's true for $2$, then it must be true for $3$, etc.)
\begin{enumerate}
\item We want to prove that $f_{S_n}(x)$ has the form above for all $n \geq 1$. That means that $n=1$ is our base case. Why do we already know this is true?
\item Next prove the induction step. In other words, \emph{assume} that for some integer $k \geq 1$ we already know that
\[
f_{S_k}(x) = \frac{\lambda e^{-\lambda x} (\lambda x)^{k-1}}{(k-1)!}.
\]
and prove that
\[
f_{S_{k+1}}(x) = \frac{\lambda e^{-\lambda x} (\lambda x)^{k}}{k!}.
\]
\emph{Hint:} $S_k = S_{k-1} + X_k$.\\
Parts (a) and (b) together prove the statement by the principle of mathematical induction.
\end{enumerate}
\item What do Chebyshev's Inequality and the Law of Large Numbers say about the probability of getting at least 75 heads when flipping a fair coin 100 times? \emph{Hint:} Improve your bound by using the fact that the binomial distribution is symmetric.
% \item Let $X$ be any random variable which takes on values $0,1,\ldots, n$ and has $\E[X] = \Var(X) = 1$. Show that, for any positive integer $k$,
% \[
% P(X \geq k + 1) \leq \frac{1}{k^2}.
% \]
\item Each student's score on a particular calculus final is a random variable with values in the range $[0,100]$, mean $70$, and variance $25$.
\begin{enumerate}
\item Find the best lower bound you can (using only the tools we've learned), for the probability that a particular student's score will fall between $65$ and $75$.
\item If $100$ students take the final, find a lower bound for the probability that the class mean will fall between $65$ and $75$.
\end{enumerate}
\item A share of common stock in the Pilsdorff beer company has price $Y_n$ on the $n$th business day of the year. ($Y_n$ is a random variable.) Finn observes that the price change $X_n = Y_{n+1} - Y_n$ appears to be a random variable with mean $\mu = 0$ and variance $\sigma^2 = 1/4$. If $Y_1 = 30$, find a lower bound for the following probabilities, under the assumption that the $X_n$'s are mutually independent.
\begin{enumerate}
\item $P(25 \leq Y_2 \leq 35)$
\item $P(25 \leq Y_{11} \leq 35)$
\item $P(25 \leq Y_{101} \leq 35)$
\end{enumerate}
\end{enumerate}
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