# Chapter 5: Discrete Probability Distributions

1. Suppose we have an experiment which consists of flipping a coin three times.

6. For each binomially-distributed random variable *x*, provide the expected value, the variance, and the standard deviation.

8. Find the binomial probability for each exercise below. Label each parameter as *x, n,* and *p*, and submit to dbinom(*x,n,p*). Hint: remember that we can answer each question by simply hitting the up arrow (to recover the dbinom() function) and entering the required *n, x,* and *p* values as needed. Any function, not only the dbinom() function, can be recycled repeatedly in this way.

9. At a large public university, half the students enrolled in a statistics class take the course on a pass-fail basis; the other half take it for a normal grade of A,...,F.

10. Suppose *x* is a Poisson-distributed random variable with an expected value of 3 occurrences per interval. Please answer the following questions.

11. A Poisson process has an expected value of 5 occurrences per interval.

14. Calls arrive at a customer-service number at the rate of 20 per hour. Use R to answer the following questions.

15. Small aircraft land at a private airport at an average rate of 1 arrival/10 minutes.

16. The number of defects associated with a plate glass manufacturing process is Poisson- distributed with a rate of 0.002 defects per square foot. Suppose the manufacturer has received an order for a large plate glass window of dimensions 10 feet by 10 feet.

17. As soon as the spring season arrives, many businesses contract for construction jobs they have been delaying over the previous cold winter months. One of the major league baseball teams has decided to resurface with asphalt a large parking area adjacent to their stadium. The dimensions of the surface are 300 feet by 200 feet, or 60,000 square feet. Unfortunately, since asphalt is usually imperfectly applied, there is an average of 0.06 air bubbles per square foot.

18. Suppose the number of radioactive particles identified by a sensitive counting device is Poisson-distributed. The particles pass through a certain threshold at a rate of 60 per millisecond. (A millisecond is one-thousandth of a second, or 1/1000 second.)

19. If *N* = 12 and *r* = 4, what are the hypergeometric probabilities* f(x)* for the following values of *x* and *n*?

Note: if *N* = 12 and *r *= 4, then (*N*– *r*) = (12 – 4) = 8

Recall: the hypergeometric probability function is

which in this case is

22. If *N* = 24, *r* = 12, and *n* = 6, what are the hypergeometric probabilities *f(x) *for the following values of x? Note: if *N* = 24 and *r* = 12, then *(N–r*) = (24–12) = 12:

25. A small manufacturer is concerned with the quality of AA batteries being delivered by a certain supplier. Recently, the average battery life has fallen below the manufacturer's standard, and so the manufacturer has begun to test the life of the batteries from this particular supplier on a regular basis. To do this, the quality control inspector randomly selects small samples of batteries from each shipping carton of 100. If even 1 of the batteries fails the test, the entire box of 100 is rejected and returned to the supplier. Suppose the current carton of 100 batteries contains 15 defective batteries.