\documentclass[12pt]{article} \usepackage[margin=1.06in]{geometry} \geometry{letterpaper} \usepackage{graphicx} \usepackage{amsmath, amssymb, amsthm} \usepackage{hyperref, multicol} \usepackage{pifont} \usepackage{marvosym} \hypersetup{colorlinks=false, allcolors=blue} \newcommand{\noin}{\noindent} \newcommand{\SD}{\textnormal{SD}} \newcommand{\var}{\textnormal{Var}} \newcommand{\Bern}{\textnormal{Bern}} \newcommand{\Bin}{\textnormal{Bin}} \newcommand{\Geom}{\textnormal{Geom}} \newcommand{\FS}{\textnormal{FS}} \newcommand{\HGeom}{\textnormal{HGeom}} \newcommand{\NBin}{\textnormal{NBin}} \newcommand{\Pois}{\textnormal{Pois}} \newcommand{\Expo}{\textnormal{Expo}} \newcommand{\Unif}{\textnormal{Unif}} \newcommand{\N}{\mathcal{N}} \begin{document} \noindent {\large \textbf{Stat 110 Homework 5, Fall 2017}} \bigskip \noindent \textbf{Due}: Friday 10/20 at 5:00 pm, submitted as a PDF via the \href{https://canvas.harvard.edu/courses/27764}{{course webpage}}. Please check carefully to make sure you upload the correct file. Your submission must be a single PDF file, no more than $20$ MB in size. It can be typeset or scanned, but must be clear and easily legible (not blurry or faint) and correctly rotated. No submissions on paper or by email will be accepted. Please show your work and give clear, careful, convincing justifications. See the syllabus for the collaboration policy. \bigskip \noindent 1. Use Poisson approximations to investigate the following types of coincidences. \medskip \noin (a) Suppose that there are $1600$ sophomores at Harvard. What is the probability that there are two Harvard sophomores who were born not only on the same day of the year, but also at the same hour \emph{and} the same minute (e.g., both sophomores were born at 2:37 pm on March 31, not necessarily in the same year)? Assume that all 365 days of the year are equally likely (and exclude February 29), all times of day are equally likely for a birth, and independence between different students' births (despite the existence of twins: \url{http://news.harvard.edu/gazette/story/2016/05/doubling-up-at-harvard/} ). \medskip \noin (b) With assumptions as in (a), what is the probability that there are \emph{four} Harvard sophomores who were born not only on the same day, but also at the same hour (e.g., all four were born between 2 pm and 3 pm on March 31, not necessarily in the same year)? Give two different Poisson approximations for this value, one based on creating an indicator r.v.~for each quadruplet of sophomores, and the other based on creating an indicator r.v.~for each possible day-hour. Which do you think is more accurate? Why? \bigskip \noindent 2. (BH 4.83) The legendary Caltech physicist Richard Feynman and two editors of \emph{The Feynman Lectures on Physics} (Michael Gottlieb and Ralph Leighton) posed the following problem about how to decide what to order at a restaurant. You plan to eat $m$ meals at a certain restaurant, where you have never eaten before. Each time, you will order one dish. The restaurant has $n$ dishes on the menu, with $n \geq m$. Assume that if you had tried all the dishes, you would have a definite ranking of them from $1$ (your least favorite) to $n$ (your favorite). If you knew which your favorite was, you would be happy to order it always (you never get tired of it). Before you've eaten at the restaurant, this ranking is completely unknown to you. After you've tried some dishes, you can rank those dishes amongst themselves, but don't know how they compare with the dishes you haven't yet tried. There is thus an \emph{exploration-exploitation tradeoff}: should you try new dishes, or should you order your favorite among the dishes you have tried before? A natural strategy is to have two phases in your series of visits to the restaurant: an \emph{exploration phase}, where you try different dishes each time, and an \emph{exploitation phase}, where you always order the best dish you obtained in the exploration phase. Let $k$ be the length of the exploration phase (so $m-k$ is the length of the exploitation phase). Your goal is to maximize the expected sum of the ranks of the dishes you eat there (the rank of a dish is the ``true" rank from $1$ to $n$ that you would give that dish if you could try all the dishes). Show that the optimal choice is $$k = \sqrt{2(m+1)}-1,$$ or this rounded up or down to an integer if needed. Do this in the following steps: \medskip \noin (a) Let $X$ be the rank of the best dish that you find in the exploration phase. Find the expected sum of the ranks of the dishes you eat, in terms of $E(X)$. \medskip \noin (b) Find the PMF of $X$, as a simple expression in terms of binomial coefficients. \medskip \noin (c) Show that $$E(X) = \frac{k(n+1)}{k+1}.$$ \smallskip \noin Hint: Use Example 1.5.2 (about the team captain) and Exercise 18 from Chapter 1 (about the hockey stick identity). \medskip \noin (d) Use calculus to find the optimal value of $k$. \bigskip \noin 3. (BH 5.1) The Rayleigh distribution from Example 5.1.7 has PDF \[f(x) = xe^{-x^2 / 2}, \quad x > 0.\] Let $X$ have the Rayleigh distribution. \medskip \noin (a) Find $P(1