\documentclass[letterpaper,11pt]{article}
\newcommand{\removefromfile}[1]{}
\usepackage{algorithm}
\usepackage[noend]{algpseudocode}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{enumitem}
\usepackage{hyperref}
\usepackage[most]{tcolorbox}
\newcommand{\expect}[1]{\mathbb{E}\left[#1\right]}
\newcommand{\Var}[1]{\mathrm{Var}\left(#1\right)}
\renewcommand{\Pr}{\mathbb{P}}
\newcommand{\Prob}[1]{\Pr\left(#1 \right)}
\begin{document}
\title{{\bf Problem Set 6} }
\author{Name: TODO}
\date{}
\maketitle
\section*{Collaborators}
TODO: List collaborators, if any.
\newpage
\section*{Task 1 (PDF)}
For this exercise, give exact answers as simplified fractions.
Define function $f_X$ by
\begin{displaymath}
f_X(x) = \left\{
\begin{array}{ll}
(1 - x^3)/2 & \mbox{if } -1 < x < 1 \\
0 & \mbox{otherwise}
\end{array}
\right. \;.
\end{displaymath}
\begin{enumerate}[label=\alph*)]
\item Show that $f_X$ has the properties required of a probability density function.
\item Compute the expectation of a random variable $X$ with $f_X$ as its PDF.
\item Compute the variance of a random variable $X$ with $f_X$ as its PDF.
\end{enumerate}
\begin{tcolorbox}
TODO: Your solution here.
\end{tcolorbox}
\newpage
\section*{Task 2 (How long?)}
Suppose that the distance from Seattle to Portland is 180 miles. You decide to buy an electric bicycle for the trip. Suppose that electric bikes have speeds that are uniformly distributed between 15 and 30 miles per hour, and you buy a random motorized bike (i.e., its speed is Uniform(15,30)).
Let $T$ be the time it takes you to ride from Seattle to Portland. What is $\expect{T}$?
Note: Recall that the indefinite integral of $g(x) = x^{-1}$ is equal to $\ln(x) + C$, assuming $x > 0$.
\begin{tcolorbox}
TODO: Your solution here.
\end{tcolorbox}
\newpage
\section*{Task 3 (Practice with uniforms)}
Suppose that $X\sim Unif(0,1)$. Let $a,b$ be constants such that $0 < a ** b | X > a} $?
\item (3 point) For any real number $y$, what is $\Prob{X \le y | X > a} $? Be sure you include all cases.
\item (6 points) Let $Y$ be a random variable such that $$F_Y(y) = \Prob{X \le y | X > a}.$$
What is $f_Y(y)$ and what kind of random variable is $Y$? (Note that when you specify a distribution from our zoo, you must specify both the name of the distribution and the values of the parameters of that distribution.)
\end{enumerate}
\begin{tcolorbox}
TODO: Your solution here.
\end{tcolorbox}
\newpage
\section*{Task 4 (Dart)}
You throw a dart at a circular target of radius $r$. Let $X$ be
the distance of your dart's hit from the center of the target. Your aim is such that $X \sim \mbox{Exponential}(2/r)$. (Note that it is possible for the dart to completely miss the target.)
\begin{enumerate}[label=\alph*)]
\item As a function of $r$, determine the value $m$ such that
$\mbox{Pr}(Xm)$. Then, for $r=4$, give the value of
$m$ to 3 decimal places.
\item What is the probability that you miss the target completely?
Give your answer to 3 decimal places.
\end{enumerate}
\begin{tcolorbox}
TODO: Your solution here.
\end{tcolorbox}
\newpage
\section*{Task 5 (Flea)}
A flea of negligible size is trapped in a large, spherical,
inflated beach ball with radius $r$. (Recall that such a ball has volume $\frac43 \pi r^3$.) At this moment, it is equally
likely to be at any point within the ball. Let $X$ be the distance
of the flea from the center of the ball. For $X$, find \ldots
\begin{enumerate}[label=\alph*)]
\item the cumulative distribution function $F_X$.
\item the probability density function $f_X$.
\item the expected value $\expect{X}$.
\item the variance $\Var{X}$
\end{enumerate}
\begin{tcolorbox}
TODO: Your solution here.
\end{tcolorbox}
\newpage
\section*{Task 6 (Kangaroos)}
The average leap of a kangaroo is 20 feet. However, because of various factors such as strength, wind, etc, the kangaroo doesn't always leap exactly 20 feet. A zoologist tells you that the kangaroo leap is normally distributed with mean 20 and variance 9.
\begin{enumerate}[label=\alph*)]
\item What is the probability that the kangaroo leaps more than 25 feet?
\item What is the probability that the kangaroo leaps between 13 and 27 feet?
\end{enumerate}
\begin{tcolorbox}
TODO: Your solution here.
\end{tcolorbox}
\newpage
\section*{Task 7 (CLT for stocks)}
Suppose that the daily price change of a certain stock on the stock market is a random variable with mean 0 and variance $\sigma^2$. Thus, if $Y_n$ is the price of the stock on the $n$-th day, then
$$Y_n = Y_{n-1} + X_n,\quad n \ge 1$$
where $X_1, X_2, \ldots$ are independent, identically distributed random variables with mean 0 and variance $\sigma^2$. Suppose also that today's stock price is 100 and $\sigma^2 = 16$. Use the Central Limit Theorem to estimate the probability that the stock price will exceed 110 after 10 days.
\begin{tcolorbox}
TODO: Your solution here.
\end{tcolorbox}
\end{document}**