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\chead{MATH 350 \hfill
Worksheet 10 \hfill Due: 2018.12.03}
\textbf{Poisson}
\begin{enumerate}
\item Flaws in a certain type of drapery material appear on average
of 1 in 150 square feet. What is the probability that at most one
flaw appears in 225 feet of drapery?
\item Suppose that 1 out of every 200 people suffer a side effect
from a certain flu vaccine. If 1000 people get the flu vaccine,
what is the probability that
\begin{enumerate}
\item At most 1 person suffers a side effect,
\item 4, 5, or 6 people suffer a a side effect?
\end{enumerate}
\end{enumerate}
\textbf{(Continuous) Uniform}
The random variable $X$ following the continuous uniform
distribution on the interval $[a, b]$, $X \sim U(a, b)$, has
probability density function
\[ f(x) = \frac{1}{b-a}.\]
\begin{enumerate}
\item Find the cumulative distribution function, $F(x)$.
\item Find the mean and variance of $X$. You decide which will be
easier, direct caclulations or using moment generating functions.
\end{enumerate}
\textbf{Exponential}
The random variable $X$ following the exponential
distribution on the interval $[0, \infty)$, $X \sim \text{Exponential}(\theta)$, has
probability density function
\[f(x) = \frac{1}{\theta} e^{-x/\theta}.\]
\begin{enumerate}
\item Find the cumulative distribution function, $F(x)$.
\item Calculate the median, the $50^{th}$ percentile, of $X$.
\item Suppose that $fone$ cell phones have a mean life time of
$1000$ days. Using the \texttt{R} function \texttt{pexp(x,
1/$\theta$)}, calculate the probability a randomly chosen $fone$
lasts longer than $1095$ days.
\end{enumerate}
\textbf{Gamma}
The random variable $X$ following the gamma
distribution (not function) on the interval $[0, \infty)$, $X \sim
\text{Gamma}(\alpha, \theta)$, has
probability density function
\[f(x) = (\Gamma(\alpha)\theta^{\alpha})^{-1} x^{\alpha - 1}
e^{-x/\theta}, \]
where $\Gamma(\cdot)$ is the gamma function, and moment generating function
\[ M(t) = (1 - t\theta)^{-\alpha} \]
\begin{enumerate}
\item Calculuate the mean and variance of $X$.
\item Suppose the number of customers per hour arriving at a store
follows a Poisson process with mean $30$. Using the \texttt{R}
function \texttt{pgamma(x, $\alpha$, 1/$\theta$)}, what is the probability
that the shopkeeper will wait more than $5$ minutes before the
second customer arrives?
\end{enumerate}
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