\begin
Discover why 18 million people worldwide trust Overleaf with their work.
\begin
Discover why 18 million people worldwide trust Overleaf with their work.
\documentclass[letterpaper,12pt,addpoints]{exam}
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\newcommand{\university}{SKIT COLLEGE }
\newcommand{\faculty}{Faculty of Applied Science and Engineering}
\newcommand{\class}{ME-201}
\newcommand{\examnum}{QUIZ \#3}
\newcommand{\content}{Discrete Random Variables \& Probability Distributions}
\newcommand{\examdate}{30 AUGUST 2019}
\newcommand{\timelimit}{50 minutes}
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\title{\Large \textbf{\university\\ \faculty\\
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\class -- \examnum \\ \content}}
\author{Instructor: Prof. CM sir}
\date{\examdate}
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\makebox[12cm]{\textbf{Name}:\ \hrulefill}
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\makebox[12cm]{\textbf{Roll Number}:\ \hrulefill}
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\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}\\
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\gradetable[v][questions]
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\clearpage
\begin{questions}
\question
In a particular game, a fair die is tossed. If the number of spots showing is either 4 or 5 you win $1, if the number of spots showing is 6 you win $4, and if the number of spots showing is 1, 2, or 3 you win nothing. Let X be the amount that you win.
Which of the following is the expected value of X?
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\question
The weight of written reports produced in a certain department has a Normal distribution with mean 60 g and standard deviation 12 g. The probability that the next report will weigh less than 45 g is
\begin{parts}
\part 0.1056
\part 0.3944
\part 0.1045
\part 0.8944
\end{parts}
\clearpage
\question A small store keeps track of the number X of customers that make a purchase during the first hour that the store is open each day. Based on the records, X has the following probability distribution.
The standard deviation of the number of customers that make a purchase during the first hour that the store is open is
\begin{parts}
\part[4] $P(X=1)$
\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}
$$P(x;\mu)=e^{-\mu}\frac{\mu^x}{x!}$$
$$E(X)=Var(X)=\mu$$
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