```
\documentclass[letterpaper,12pt,addpoints]{exam}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage[top=1in, bottom=1in, left=0.75in, right=0.75in]{geometry}
\usepackage{amsmath,amssymb}
\newcommand{\university}{UNIVERSITY OF TORONTO}
\newcommand{\faculty}{Faculty of Applied Science and Engineering}
\newcommand{\class}{CME263H1}
\newcommand{\examnum}{QUIZ \#3}
\newcommand{\content}{Discrete Random Variables \& Probability Distributions}
\newcommand{\examdate}{2018/02/15}
\newcommand{\timelimit}{50 minutes}
\pagestyle{headandfoot}
\firstpageheader{}{}{}
\firstpagefooter{}{Page \thepage\ of \numpages}{}
\runningheader{\class}{\examnum}{\examdate}
\runningheadrule
\runningfooter{}{Page \thepage\ of \numpages}{}
\begin{document}
\title{\Large \textbf{\university\\ \faculty\\
\bigskip
\class -- \examnum \\ \content}}
\author{Instructor: Prof. Amer Shalaby}
\date{\examdate}
\maketitle
\begin{flushleft}
\makebox[12cm]{\textbf{Name}:\ \hrulefill}
\medskip
\makebox[12cm]{\textbf{Student Number}:\ \hrulefill}
\end{flushleft}
\noindent \rule{\textwidth}{1pt}
\noindent This exam contains \numpages\ pages (including this cover page) and \numquestions\ questions. Total of points is \numpoints.\\
Good luck and Happy reading work!
\begin{center}
\textbf{Distribution of Marks}\\
\medskip
\gradetable[v][questions]
\end{center}
\clearpage
\begin{questions}
\question
A contractor is required by a county planning department to submit one, two, three, four, or five forms (depending on the nature of the project) in applying for a building permit. Let $Y=$ the number of forms required of the next applicant. The probability that $y$ forms are required is known to be proportional to $y$---that is, $p(y)=ky\ \text{for}\ y=1,\dots,5.$
\begin{parts}
\part[2] What is the value of $k$? (\textit{Hint}: $\sum_{y=1}^5p(y)=1$)
\part[2] What is the probability that at most three forms are required?
\part[2] What is the probability that between two and four forms (inclusive)are required?
\part[2] Could $p(y)=y^2/50\ \text{for}\ y=1,\dots,5$ be the \textit{pmf} of $Y$.
\end{parts}
\clearpage
\question
The \textit{pmf} of the amount of memory \textit{X} (GB) in a purchased flash drive was given as
\begin{center}
\begin{tabular}{c|c|c|c|c|c}
$x$ & 1 & 2 & 4 & 8 & 16 \\
\hline
$p(x)$ & 0.05 & 0.10 & 0.35 & 0.40 & 0.10\\
\end{tabular}
\end{center}
Compute the following:
\begin{parts}
\part[2] Expected value $E(X)$
\part[2] Variance $V(X)$ directly from the definition
\part[2] The standard deviation $\sigma(X)$
\part[2] $V(X)$ using the shortcut formula ($V(X)=E(X^2)-E^2(X)$)
\end{parts}
\clearpage
\question Each of 12 refrigerators of a certain type has been returned to a distributor because of the presence of a high-pitched oscillating noise when the refrigerator is running. Suppose that 5 of these 12 have defective compressors and the other 7 have less serious problems. If they are examined in random order, let X = the number among the first 6 examined that have a defective compressor. Compute the following:
\begin{parts}
\part[3] $P(X=1)$
\part[3] $P(X \geq 4)$
\end{parts}
\clearpage
\question A reservation service employs five information operators who receive requests for information independently of one another, each according to a Poisson process with rate $\mu=2$ per minute.
\begin{parts}
\part[4] What is the probability that during a given 1-min period, the first operator receives no requests?
\part[4] What is the probability that during a given 1-min period, exactly four of the five operators receive no requests?(\textit{Hint}: treat either as a binomial process of 5 trials with 4 successes or consider 5 combinations of Poisson processes, e.g. only 1st operation receives a request or only 2nd operation receives a request and so on)
\end{parts}
\end{questions}
\clearpage
\centering \textbf{\large Probability mass/distribution functions}
\flushleft \textbf{Binomial Distribution}
$$f(x;n,p)=b(x;np)=\binom{n}{x}p^x(1-p)^{n-x}$$
$$\mu=E(x)=np$$
$$\sigma^2_x=np(1-p)$$
\flushleft \textbf{Hypergeometric Distribution}
$$P(X=x)=h(x;n,M,N)=\frac{\binom{M}{x}\binom{N-M}{n-x}}{\binom{N}{n}}$$
$$\mu=E(X)=\frac{nM}{N}$$
$$\sigma^2_x=n\frac{M}{N}\frac{N-M}{N}\frac{N-n}{N-1}$$
\flushleft \textbf{Poisson Distribution}
$$P(x;\mu)=e^{-\mu}\frac{\mu^x}{x!}$$
$$E(X)=Var(X)=\mu$$
\clearpage
This page is intentionally left blank to accommodate work that wouldn't fit elsewhere and/or scratch work.
\end{document}
```