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\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{I'm allergic to polling [6 points]}
    
    In lecture, we discussed an {\textbf idealized} polling procedure and analysis to estimate the fraction $p$ of a population that loves cats.
    \newline\newline
    Specifically, we analyzed how we could choose the number $n$ of people to sample in order to guarantee  that 95\% of the time, it will be the case that $\hat{p} = \bar{X} \in [p- .05, p + 0.05].$
    \newline\newline
    You probably know that many recent polls (that follow similar methodology) have been \textbf {way off}. Briefly discuss \textbf{two} reasons that real-world polling may not work as well as the idealized polling that we discussed.
    \newline\newline
    Note: There can be many different answers. We expect most reasons will be 1-2 sentences.

    \begin{tcolorbox}
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    TODO: Your solution here
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    \newpage
    \section{Please I'm a Star [15 points]}
    An astronomer would like to measure the distance (in light years) from her observatory to a distant star. Each measurement is noisy, yielding only an estimate of the distance. Therefore, the astronomer plans to make a series of independent measurements and then use the average value of these measurements as her estimate of the actual distance. Suppose that each measurement has mean $D$ (the true distance) and a variance of $7$ light years. How many measurements does she need to make to be 96\% confident that her estimate is accurate to within $\pm 0.3$ light years? Your solution should use the CLT.

    \begin{tcolorbox}
    \begin{enumerate}
    TODO: Your solution here
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    \newpage
    \section{Take me to a Higher Level [25 points]}
    You watch dramas for $H$ hours, where $H$ is a random variable, equally likely to be 1, 2 or 3. The number of episodes $E$ that you watch is random and depends on how long you watch for. (The episodes are of different lengths...). We are told that
    $$\mathbb{P}(E = e \mid H = h) = \frac{1}{h},\quad\quad \text{ for }e=1, \ldots, h \;.$$
    
    
    \begin{enumerate}[label=(\alph*)]
    \item (5 points) Find the joint distribution of $E$ and $H$.
    \item (5 points) Find the marginal distribution of $E$.
    \item (5 points) Find the conditional distribution of $H$ given that $E=1$ (that is, $\mathbb{P}(H= h \mid E=1)$ for each possible $h$ in 1,2,3). Use the definition of conditional probability and the results from previous parts.
    \item (10 points) Suppose that you watched 1 or 2 episodes. Find the expected number of hours you watched conditioned on this event, defined as follows:
    $$\mathbb{E}[H \mid E=1 \cup E=2] = \sum_{h=1}^3 h \cdot \mathbb{P}(H=h \mid E=1 \cup E=2)$$
    \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
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    \newpage
    \section{Joint Density Ventures [12 points]}

    Let $X$ and $Y$ be continuous random variables with the following joint distribution.
    
    \begin{equation*}
    f_{X,Y}(x,y) =
        \begin{cases}
            \frac{3}{2}x^2y & 0 \leq x \leq 1, 0 \leq y \leq 2\\
            0 & \text{otherwise}
        \end{cases}
    \end{equation*}

    \begin{enumerate}[label=(\alph*)]
    \item Find $\mathbb{P}(X > Y)$ [5 points].
    \item What is the marginal PDF of X? [3 points]
    \item What is the marginal PDF of Y? [3 points]
    \item Are $X$ and $Y$ independent? [1 point]
    \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
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    \newpage
    \section{Don't Dimension It [10 points]}
    Let $X,Y,Z$ be independent and uniformly distributed over (0,1) Compute $$\mathbb{P}(X \ge YZ).$$
    Your answer should be a number.
    \newline\newline
    Hint: You just need to integrate over the relevant region of the joint density $f_{X,Y,Z}(x,y,z)$.

    \begin{tcolorbox}
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    TODO: Your solution here
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    \newpage
    \section{Law of Expected Employment [15 points]}
    12 students have decided to apply for a TA position next quarter.
    The number of courses that they can be a TA for is a Poisson random variable with mean 5. Suppose that each student independently chooses exactly one of the courses to apply for uniformly at random (and independently of the choices of the other applicants). Suppose also that each student has probability 0.2 of being acceptable as a TA to the professor teaching any course they apply for.  Use the law of total expectation to compute the expected number of courses that have at least one applicant acceptable to the professor for that course. Your final answer should have only one summation symbol.

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    TODO: Your solution here
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\end{document}