Statistics hw3

hw3.txt

3-86: The number of people arriving for treatment at an emergency room can be modeled by a Poisson process with a rate parameter of five per hour.
	a) What is the probability that exactly four arrivals occur during a particular hour?
	b) What is the probability that at least four people arrive during a particular hour?
	c) How many people do you expect to arrive during a 45-min period?
3-90: Let $X$ have a Poisson distribution with parameter $\mu$. Show that $E(X) = \mu$ directly from the definition of expected value. [\emph{Hint:} The first term in the sum equals 0, and then $x$ can be canceled. Now factor out $\mu$ and show that what is left sums to 1.
4-4: Let $X$ denote the vibratory stress (psi) on a wind turbine blade at a particular wind speed in a wind tunnel. The article "Blade Fatigue Life Assessment with Application to VAWTS" (\emph{J. of Solar Energy Engr.}, 1982: 107-111) proposes the Rayleigh distribution, with pdf
    \[ f(x; \theta) = \left\{ \begin{matrix} \frac{x}{\theta^2} \cdot e^{-x^2/(2\theta^2)} & x > 0 \\ 
    0 & \textrm{otherwise} \end{matrix} \right. \]
	as a model for the $X$ distribution.
	a) Verify that $f(x; \theta)$ is a legitimate pdf.
	b) Suppose $\theta = 100$ (a value suggested by a graph in the article). What is the probability that $X$ is at most 200? Less than 200? At least 200?
	c) What is the probability that $X$ is between 100 and 200 (again assuming $\theta = 100$)?
	d) Give an expression for $P(X \leq x)$.
4-20: Consider the pdf for total waiting time $Y$ for two buses
    \[ f(y) = \left\{ \begin{matrix} \frac{1}{25}y && 0 \leq y < 5 \\
                             \frac{2}{5} - \frac{1}{25}y && 5 \leq y < 10 \\
                             0 & \textrm{otherwise}
                      \end{matrix} \right. \]
	introduced in Exercise 8.
	a) Compute and sketch the cdf of $Y$. [\emph{Hint:} Consider separately $0 \leq y < 5$ and $5 \leq y < 10$ in computing $F(y)$. A graph of the pdf should be helpful.]
	b) Obtain an expression for the $(100p)$th percentile. [\emph{Hint:} Consider separately $0 < p < .5$ and $.5 < p < 1$]
	c) Compute $E(Y)$ and $V(Y)$. How do these compare with the expected waiting time and variance for a single bus when the time is uniformly distributed on [0, 5]?
4-28: Let $Z$ be a standard normal random variable and calculate the following probabilities, drawing pictures wherever appropriate.
	a) $P(0 \leq Z \leq 2.17)$
	b) $P(0 \leq Z \leq 1)$
	c) $P(-2.50 \leq Z \leq 0)$
	d) $P(-2.50 \leq Z \leq 2.50)$
	e) $P(Z \leq 1.37)$
	f) $P(-1.75 \leq Z)$
	g) $P(01.50 \leq Z \leq 2.00)$
	h) $P(1.37 \leq Z \leq 2.5)$
	i) $P(1.50 \leq Z)$
	j) $P(|Z| \leq 2.50)$
4-30: Find the following percentiles for the standard normal distribution. Interpolate wherever appropriate.
	a) 91st
	b) 9th
	c) 75th
	d) 25th
	e) 6th
4-32: Suppose the force acting on a column that helps to support a building is a normally distributed random variable $X$ with mean value 15.0 kips and standard deviation 1.25 kips. Compute the following probabilities by standardizing and then using table A.3.
	a) $P(X \leq 15)$
	b) $P(X \leq 17.5)$
	c) $P(X \geq 10)$
	d) $P(14 \leq X \leq 18)$
	e) $P(|X - 15| \leq 3)$
4-40: The article "Monte Carlo Simulation--Tool for Better Understanding of LRFD" (\emph{J. of Structural Engr.}, 1993: 1586-1599) suggests that yield strength (ksi) for A36 grade steel is normally distributed with $\mu = 43$ and $\sigma = 4.5$.
	a) What is the probability that yield strength is at most 40? Greater than 60?
	b) What yield strength value separated the strongest 75\% from the others?
4-70: If $X$ has an exponential distribution with parameter $\lambda$, derive a general expression for the $(100p)$th percentile of the distribution. Then specialize to obtain the median.