Suppose that trees are distributed in a forest according to a two-dimensional Poisson process with parameter $\alpha$, the expected number of trees per acre, equal to 80. a. What is the probability that in a certain quarter-acre plot, there will be at most 16 trees? b. If the forest covers 85,000 acres, what is the expected number of trees in the forest? c. Suppose you select a point in the forest and construct a circle of radius .1 mile. Let X=the number of trees within that circular region. What is the pmf of X? [Hint: 1 sq mile=640 acres.]

In proof testing of circuit boards, the probability that any particular diode will fail is .01. Suppose a circuit board contains 200 diodes. a. How many diodes would you expect to fail, and what is the standard deviation of the number that are expected to fail? b. What is the (approximate) probability that at least four diodes will fail on a randomly selected board? c. If five boards are shipped to a particular customer, how likely is it that at least four of them will work properly? (A board works properly only if all its diodes work.)

Differentiate the functions with respect to the independent variable. $g(x)=2^{2 \cos x}$

Let X have a Poisson distribution with parameter $\lambda$. Show that $E(X)=\lambda$ directly from the definition of expected value. [Hint: The first term in the sum equals 0, and then x can be canceled. Now factor out $\lambda$ and show that what is left sums to 1.]

Find the derivative of $f(x)=\tan ^{3}\left(3 x^{3}-3\right)$

Suppose small aircraft arrive at a certain airport according to a Poisson process with rate $\alpha =8$ per hour, so that the number of arrivals during a time period of t hours is a Poisson rv with parameter $\lambda=8 t$. a. What is the probability that exactly 6 small aircraft arrive during a 1-hour period? At least 6? At least 10? b. What are the expected value and standard deviation of the number of small aircraft that arrive during a 90-min period? c. What is the probability that at least 20 small aircraft arrive during a $2 \frac{1}{2}$-hour period? That at most 10 arrive during this period?

The number of requests for assistance received by a towing service is a Poisson process with rate $\alpha =4$ per hour. a. Compute the probability that exactly ten requests are received during a particular 2-hour period. b. If the operators of the towing service take a 30-min break for lunch, what is the probability that they do not miss any calls for assistance? c. How many calls would you expect during their break?

Let X be the number of material anomalies occurring in a particular region of an aircraft gas-turbine disk. The article “Methodology for Probabilistic Life Prediction of Multiple-Anomaly Materials” (Amer. Inst. of Aeronautics and Astronautics J., 2006: 787–793) proposes a Poisson distribution for X. Suppose that $\mu=4$. a. Compute both $P(X \leq 4)$ and $P(X<4)$. b. Compute $P(4 \leq X \leq 8)$. c. Compute $P(8 \leq X)$. d. What is the probability that the number of anomalies exceeds its mean value by no more than one standard deviation?

Compute $f(c+h)-f(c)$ at the indicated point. $f(x)=3 x^{2} ; c=1$

In proof testing of circuit boards, the probability that any particular diode will fail is .01. Suppose a circuit board contains 200 diodes. a. How many diodes would you expect to fail, and what is the standard deviation of the number that are expected to fail? b. What is the (approximate) probability that at least four diodes will fail on a randomly selected board? c. If five boards are shipped to a particular customer, how likely is it that at least four of them will work properly? (A board works properly only if all its diodes work.)

Suppose that the number of drivers who travel between a particular origin and destination during a designated time period has a Poisson distribution with parameter $\lambda=20$ (suggested in the article “Dynamic Ride Sharing: Theory and Practice,” J. of Transp. Engr., 1997: 308-312). What is the probability that the number of drivers will a. Be at most 10? b. Exceed 20? c. Be between 10 and 20, inclusive? Be strictly between 10 and 20? d. Be within 2 standard deviations of the mean value?

Suppose that only .10 % of all computers of a certain type experience CPU failure during the warranty period. Consider a sample of 10,000 computers. a. What are the expected value and standard deviation of the number of computers in the sample that have the defect? b. What is the (approximate) probability that more than 10 sampled computers have the defect? c. What is the (approximate) probability that no sampled computers have the defect?

Sketch the graph of a continuous function on $[0, \infty)$ with f(0) = 0 and $\lim _{x \rightarrow \infty} f(x)=1.$

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?