\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{Viva la correlation [15 Points]}

    Recall the `Hungry Washing Machine' question from section. You have $10$ pairs of socks, where each is a different color. Left and right socks are distinct. Your washing machine randomly eats 4 socks. Suppose that your washing machine is equally likely to eat any 4 of your 20 distinct socks. For $i=1,\dots,10$, let $X_i$ be the indicator/Bernoulli random variable that is $1$ if pair $i$ is complete after this fiasco and $0$ otherwise.
    Let $Y=\sum_{i=1}^{10}{X_i}$ be the \textit{total} number of complete pairs of socks that you have left.
    
    \begin{enumerate}[label=(\alph*)]
        \item What is $Cov(X_i, X_j)$ where $1 \leq i,j \leq 10$ and $i = j$? [4 points]
    
        \item What is $Cov(X_i, X_j)$ where $1 \leq i,j \leq 10$ and $i \neq j$? [4 points]
        
        \item What is $Var(Y)$? \textbf{Give your answer as a simplified fraction}. Hint: You will want to use your answers to the previous parts. [7 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{My name is Jared, I'm 19 [16 points]}
    You read a book, consisting of $50$ chapters. It takes an average of $2$ hours to read a chapter, with a variance of $0.2$ hours. The time to read each chapter is independent.
    \newline \newline
    You should treat time as continuous for this problem.

    \begin{enumerate}[label=(\alph*)]
        \item What is the expectation of the total time to read the book? [2 points]
        \item What is the variance of the total time to read the book? [2 points]
        \item You have $150$ hours
        %250 hours 
        to read between now and when you need to finish the book for your book club. Use Markov's Inequality to bound the probability that you finishes the book before your club meets. Hint: you'll need to take a complement. [6 points]
        \item You will have 120
        %115 
        hours to read between now and the start of your book club meetings. Use Chebyshev's inequality to bound the probability that you finish the book after at least $80$ hours, but in time for your book club meetings. [6 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{We continue to yearn [15 points]}

    You have 2500 urns, that you place into a 50-by-50 grid. You will throw \textbf{40,000} balls (independently) toward the grid of urns, with equal probability for the ball to go in each urn.
    \newline\newline
    You hope that at the end of the process, each urn will have at least $4$ balls inside. You want to upperbound the probability that you will fail to get at least $4$ balls into every urn.
        
    \begin{enumerate}[label=(\alph*)]
        \item Use a Chernoff bound from class to bound the probability that the urn in the lower-left corner of the grid does not have at least $4$ balls inside.  [7 points]
        \item Is the probability of the lower-left urn having less than $4$ balls independent of the probability that the urn in the upper-right corner has less than $4$ balls? Briefly explain (you may give a formal derivation/calculation as an explanation or an informal one). [3 points]
        \item Bound the probability that at least one urn has fewer than $4$ balls. Give the best bound you can from your answers in (a) and (b). The bound may be trivial; sometimes, concentration inequalities are uninformative. [5 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{Gotta Estimate 'em All! [15 points]} 
    Every 312 student is surveyed on their preferred starter Pokemon between Bulbasaur, Charmander, and Squirtle. Suppose that each student independently chooses Bulbasaur with probability $\theta_B$, Charmander with probability $\theta_C$ and Squirtle with probability $1 - \theta_B - \theta_C$. (Thus, $0 \leq \theta_B + \theta_C \leq 1).$ The parameters $\theta_B, \theta_C$ are unknown. Suppose that $x_1, ..., x_n$ are $n$ independent, identically distributed samples from this distribution. Let $n_B$ be the number of $x_i$'s equal to Bulbasaur, let $n_C $ be the number of $x_i$'s equal to Charmander and let $n_S$ be the number of $x_i$'s equal to Squirtle. What are the maximum likelihood estimates for $\theta_B$ and $\theta_C$ in terms of $n_B, n_C,$ and $n_S$?
    \newline \newline 
     In doing this problem, you do not need to do a second-derivative test (or any other test) to confirm you have a maximizer. You may assume any critical point you find is a maximizer.
     \newline \newline
     \textbf{For this problem, except where noted, all of your work must be understandable without reference to any calculator (including Wolfram-Alpha). You may check your answers using calculators, but your explanation may not rely on them.}
 
    \begin{enumerate} [label=(\alph*)]
        \item Write the Likelihood function. [3 points]
        \item Write the log-likelihood. [2 points]
        \item Give two equations describing the maximizer of the log-likelihood. Be sure to show work (e.g., taking of partial derivatives) on how you got these equations. [8 points]
    
        \item You should now have a system of two (linear) equations in two unknowns. Find the solution to this system, and use it to write the MLEs for $\theta_B$ and $\theta_C$. You do \textbf{not} have to show your work for this part, and may use online algebra solvers if you wish. [2 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{To be continued [24 points]}
    You do not need to check second order conditions in the following.

    \begin{enumerate}[label=(\alph*)]
        \item Let $x_1, x_2, \ldots, x_n$ be independent
      samples from an exponential distribution with unknown parameter
      $\lambda$.  What is the maximum likelihood estimator for $\lambda$? [12 Points]
    
        \item Given $\theta>0$. Suppose that $x_1, \ldots, x_n$ are i.i.d. realizations (aka samples) from the model
        $$f(x; \theta) = \begin{cases}
        \theta x^{\theta -1} & 0 < x < 1 \\
        0 & \text{otherwise}.
        \end{cases}
        $$      
    \end{enumerate}
    Find the maximum likelihood estimate for $\theta$. [12 Points] 
    \begin{tcolorbox}
    \begin{enumerate} [label=(\alph*)]
            \item TODO: Your solution here
            \item TODO: Your solution here
    \end{enumerate}
    \end{tcolorbox}

    \newpage
    \section{A journey, not the estimation [5 points]}
     Let $X_1, X_2, ..., X_n$ be random samples drawn from the continuous distribution $\text{Uniform}(0, \theta)$, for $\theta > 0$. We stumble upon a possible estimator $\hat{\theta}$ for $\theta$, where  $\hat{\theta} = 2 \cdot \frac{\sum\limits_{i=1}^n X_i}{n}$. Is $\hat{\theta}$ an unbiased estimator of $\theta$?

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