\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 6, Fall 2017}} \bigskip \noindent \textbf{Due}: Friday 10/27 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. (BH 5.47) Emails arrive in an inbox according to a Poisson process with rate $20$ emails per hour. Let $T$ be the time at which the $3$rd email arrives, measured in hours after a certain fixed starting time. Find $P(T > 0.1)$ without using calculus. \bigskip \noindent 2. (BH 6.16) For any r.v.~$X$ with mean $\mu$ and variance $\sigma^2$, we define the \emph{skewness} of $X$ to be the third standardized moment of $X$: $$\textrm{Skew}(X) = E \left( \frac{X-\mu}{\sigma} \right)^3.$$ Let $X \sim \Expo(\lambda)$. Find the skewness of $X$, and explain why it is positive and why it does not depend on $\lambda$. \smallskip \noin Hint: Recall that $\lambda X \sim \Expo(1)$ and the $n$th moment of an $\Expo(1)$ r.v.~is $n!$ for all $n$. \bigskip \noindent 3. (BH 6.19) Use MGFs to determine whether $X+2Y$ is Poisson, for $X,Y$ i.i.d.~$\Pois(\lambda)$. \bigskip \noindent 4. (BH 6.25) Let $Y=X^\beta$, with $X \sim \Expo(1)$ and $\beta > 0$. The distribution of $Y$ is called the \emph{Weibull}\index{Weibull distribution} distribution with parameter $\beta$. This generalizes the Exponential, allowing for non-constant hazard functions. Weibull distributions are widely used in statistics, engineering, and survival analysis; there is even an 800-page book devoted to this distribution: \emph{The Weibull Distribution: A Handbook} by Horst Rinne. \medskip \noin For this problem, let $\beta=3$. \medskip \noin (a) Find $P(Y > s + t | Y > s)$ for $s,t>0$. Does $Y$ have the memoryless property? \medskip \noin (b) Find the mean and variance of $Y$, and the $n$th moment $E(Y^n)$ for $n=1,2,\dots$. \medskip \noin (c) Determine whether or not the MGF of $Y$ exists. \bigskip \noindent 5. (BH 7.1) Alice and Bob arrange to meet for lunch on a certain day at noon. However, neither is known for punctuality. They both arrive independently at uniformly distributed times between noon and 1 pm on that day. Each is willing to wait up to 15 minutes for the other to show up. What is the probability they will meet for lunch that day? \bigskip \noindent 6. (BH 7.2) Alice, Bob, and Carl arrange to meet for lunch on a certain day. They arrive independently at uniformly distributed times between 1 pm and 1:30 pm on that day. \medskip \noin (a) What is the probability that Carl arrives first? \medskip \noin \emph{For the rest of this problem, assume that Carl arrives first at 1:10 pm, and condition on this fact.} \medskip \noin (b) What is the probability that Carl will be waiting alone for more than 10 minutes? \medskip \noin (c) What is the probability that Carl will have to wait more than 10 minutes until his party is complete? \medskip \noin (d) What is the probability that the person who arrives second will have to wait more than $5$ minutes for the third person to show up? \bigskip \noindent 7. (BH 7.7) A stick of length $L$ (a positive constant) is broken at a uniformly random point $X$. Given that $X=x$, another breakpoint $Y$ is chosen uniformly on the interval $[0,x]$. \medskip \noin (a) Find the joint PDF of $X$ and $Y$. Be sure to specify the support. \medskip \noin (b) We already know that the marginal distribution of $X$ is $\Unif(0,L)$. Check that marginalizing out $Y$ from the joint PDF agrees that this is the marginal distribution of $X$. \medskip \noin (c) We already know that the conditional distribution of $Y$ given $X=x$ is $\Unif(0,x)$. Check that using the definition of conditional PDFs (in terms of joint and marginal PDFs) agrees that this is the conditional distribution of $Y$ given $X=x$. \medskip \noin (d) Find the marginal PDF of $Y$. \medskip \noin (e) Find the conditional PDF of $X$ given $Y=y$. \end{document}