\documentclass[11pt, letter]{article}
\usepackage[inner=1.5cm,outer=1.5cm,top=2.5cm,bottom=2.5cm]{geometry}
\usepackage{enumitem}
\usepackage[most]{tcolorbox}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{hyperref}
\setcounter{section}{-1} 

\begin{document}
    \textbf{Submission instructions are in the homework handout. The handout is the final source of truth for instructions and question statements over this template file, which exists only for your convenience.}

    \section{Collaborators}
    \begin{tcolorbox}
        TODO: Add collaborators here.
    \end{tcolorbox}

    \newpage
    \section{PDF or DOCX? [18 points]}
    Consider the following PDF:
    \[ f_X(x) = \begin{cases}
    r(1 - 4x^3) &\text{if}~ -\frac{1}{3} \leq x \leq \frac{1}{3} \\
    0 &\text{otherwise}
    \end{cases}
    \]
    \newline
    Where $r$ is a constant.
    
    \begin{enumerate}[label=(\alph*)]
        \item What value of $r$ makes the PDF valid? [5 points]
        \item What is $F_X(k)$? [5 points]
        \item What is $\mathbb{E}[X]$? [4 points]
        \item What is $Var(X)$? [4 points]
    \end{enumerate}
    \begin{tcolorbox}
    
    \begin{enumerate}[label=(\alph*)]
        \item     TODO: Your solution here

        \item     TODO: Your solution here

        \item     TODO: Your solution here

        \item     TODO: Your solution here
    \end{enumerate}
\end{tcolorbox}

    \newpage
        \section{Fleas get Degrees [18 points]}
    A flea of negligible size is trapped in a large, spherical,
      inflated beach ball with radius $c$.  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 

      \begin{enumerate}[label=(\alph*)]
    \item the cumulative distribution function $F$. [5 points]
    \item the probability density function $f$. [5 points]
    \item the expected value. [4 points]
    \item the variance. [4 points]
    
    Reminder: the volume of a sphere of radius $r$ is $\frac{4}{3} \pi r^3$.
      \end{enumerate}

    \begin{tcolorbox}
    \begin{enumerate}[label=(\alph*)]
        \item     TODO: Your solution here

        \item     TODO: Your solution here

        \item     TODO: Your solution here

        \item     TODO: Your solution here
    \end{enumerate}
    \end{tcolorbox}

      \newpage
      \section{A Penny for Your Darts [12 points]}
    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.  You
      have improved and your aim is such that $X \sim \mbox{Exponential}(7/r)$.  (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
      $\mathbb{P}(X<m)=\mathbb{P}(X>m)$. [6 points] 

     
    \item  For $r=15$, give the value of
      $m$ to 3 decimal places. [2 points]

    \item What is the probability that you miss the target completely?
      Give your answer to 4 decimal places. [4 points]

      \end{enumerate}
    

    \begin{tcolorbox}
    \begin{enumerate}[label=(\alph*)]
        \item     TODO: Your solution here

        \item     TODO: Your solution here

        \item     TODO: Your solution here

    \end{enumerate}
    \end{tcolorbox}

    \newpage
    \section{I'm So $\frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$ [12 points]}
    For each question below, if you need to use the CDF of a normal, be sure to round the $z$-score to the hundredths-place and use the table linked on the webpage.

    You fold a paper airplane whose flight length (in feet) is normally distributed with mean $6$ and variance $9$.
    
    \begin{enumerate}[label=(\alph*)]
    \item What is the probability that your airplane flies over 10 feet? [4 points]

    \item What is the probability that your airplane flies between 3 and 14 feet?  [4 points]

    \item Your friend folds a paper airplane whose flight length is also normally distributed, but with mean $8$ and variance $16$. What is the probability that in an airplane competition, your airplane flies further than your friend's?  [4 points]
    \end{enumerate}

        \begin{tcolorbox}
    \begin{enumerate}[label=(\alph*)]
        \item     TODO: Your solution here

        \item     TODO: Your solution here

        \item     TODO: Your solution here

    \end{enumerate}
    \end{tcolorbox}

    \newpage
\section{Wonderful Form of the Cosmic Order [24 points]}

    In all of the following use the Central Limit Theorem. You should determine if the continuity correction is necessary, and use it when it is.

    \begin{enumerate}[label=(\alph*)]
        \item You write a new sorting algorithm. The runtime for this algorithm averages $\frac{35}{3}$ seconds, with a standard deviation of 2 seconds. What is the probability that it takes less than 100 seconds to run the algorithm 9 times, with each run independent from the other? Give your answer to 4 decimal places. [8 points]

        \item You open a new bookstore. The average price of each book is $\$17$, and the prices are continuously distributed with a variance of $\$15$. You sell 40 books in a day.

        Your rich BFF offers to lend you money for your venture, on the condition that you answer the following question: at least how much money will you make at least $95\%$ of the time? Give your answer to 2 decimal places. [8 points]

        \item You play roulette, with probability $\frac{17}{35}$ of winning on a spin. One fine day, you decide to spend your afternoon gambling, and spin the wheel $n$ times. At least how big should $n$ be to guarantee you will win at least $10$ times with probability $90\%$? [8 points]
    \end{enumerate}

    \begin{tcolorbox}
    \begin{enumerate}[label=(\alph*)]
        \item     TODO: Your solution here

        \item     TODO: Your solution here

        \item     TODO: Your solution here

    \end{enumerate}
    \end{tcolorbox}

    \newpage
    \section{Practice with uniforms [12 points]}
Suppose that $X\sim Unif(0,1)$. Let $a,b$ be constants such that  $0 < a <b < 1$.

\begin{enumerate}[label=(\alph*)]
    \item (3 points) What is $\mathbb{P}(X > b | X > a)$?


    
    \item (3 point) For any real number $y$, what is $\mathbb{P}(X \le y | X > a) $? Be sure you include all cases.


    
    \item (6 points) Let $Y$ be a random variable such that $$F_Y(y) = \mathbb{P}(X \le y | X > a).$$
    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}
    \begin{enumerate}[label=(\alph*)]
        \item     TODO: Your solution here

        \item     TODO: Your solution here

        \item     TODO: Your solution here

    \end{enumerate}
    \end{tcolorbox}

\newpage
\section{Rolling in the Dice [12 points]}
Suppose that a fair 6-sided die is rolled repeatedly, with each roll independent of the others.
Let $Z$ be the number of rolls until (and including) the first time either a 2 or a 3 is rolled,
and let $W$ be the number of 6's rolled until the first 2 or 3 is rolled.
So, for example if the sequence of die values until the first 2 or 3 is 1,5,4,4,5,6,1,6,2,
then $Z$ is 9 and $W$ is 2.
\newline\newline
Define $$p(j) := \begin{cases}
\Pr(W = j \mid Z = i)& j \in \{0, 1, \ldots, i-1\}\\
0 & \text{otherwise}
\end{cases}$$
Show that $p(j)$ is the probability mass function of a binomially distributed random variable and determine its parameters $n$ and $p$.

\begin{tcolorbox}
TODO: Your solution here
    \end{tcolorbox}

\end{document}