Homework 5, Math 320, Spring 2001

Name:____________________________      Section number:______

## Math 320 Homework #5 --- Due 2/23

Include your name, section number, and homework number on every page that you hand in. Enter ``Section 1'' for the morning class (10-11AM) and ``Section 2'' for Professor Sawyer's class (12-1PM).

You can begin the exposition of your work on this page. If more room is needed, continue on sheets of paper of exactly the same size (8.5 x 11 inches), lined or not as you wish, but not torn from a spiral notebook. You should do your initial work and calculations on a separate sheet of paper before you write up the results to hand in.

Output from Excel must have your name, Section number, and the homework number in cell A1. Staple all sheets together. No unstapled papers will be accepted.

1. (i) Let X be a random variable taking the values 1,2,3,4,5 with probability function f(1) = .30,  f(2) = .15,  f(3) = 0.10,  f(4) = .15,  and f(5) = .30.  Draw a graph of the probability function. How does it compare with the graph in Figure 5.34? (Compare with exercise 5.40 on page 219.)

(ii) Consider two independent random variables X1, X2 that have the same probability function in problem 1 above. Let Y=X1+X2 . Thus Y takes values between 2 and 10 . Graph the probability function of Y . How does it compare with the graph in part (i)?
(Hint: Since X1 and X2 are independent, P(Y=y)= Suma=15 P(X1=a and X2=y-a) = Suma=1a=y-1 P(X1=a) P(X2=y-a) = Suma=1a=y-1 f(a)f(y-a) . Compare with figures 5.34, 5.35, 5,36, and 5.37.)

2. The waiting time in minutes between calls to an emergency service in a medium-sized city for 15 calls was

16, 152, 70, 40, 93, 111, 3, 10, 50, 155, 46, 166, 43, 130, and 17 .
(i) Estimate the mean time between incoming calls.
(ii) Assuming that the time between calls has an exponential distribution, estimate the rate constant r or lambda. Using the estimated value of the rate constant or lambda, find the probability that the next call will arrive in 15 minutes or less. Also estimate the probability that the next call will not arrive for 140 minutes or more.

3. Do exercise 5.60 on page 231.

4. The number of visitors X in an evening to a particular hospital emergency room has a Poisson distribution with mean mu=100. Find the probability that there will be 115 or more calls in a particular evening in two different ways:

(i) Use a computer to find P(X>=115) exactly, for example by using `poissoncdf` on a TI-83 calculator or by using the POISSON function on the Statistical function menu in Excel.
(ii) Approximate X by a normal variable Y where X and Y have the same mean and standard deviation and find P(Y>=115-0.50). (This is a continuity correction. Like with the binomial distribution, it allows for the fact that P(X=115)>0 for the Poisson but P(Y=115)=0 for the normal.)
How close are the two answers?

5. (i) Assume that X is a normal random variable with mean mu=1 and standard deviation sigma=2. Find P(X>=2.5). What is the corresponding Z score?

(ii) Let X1, X2, X3, ..., X16 be an independent sample of n=16 normal random variables, where each has the same mean and standard deviation as in part (i).  Let Xbar be the sample mean of the 16 random variables. What is the mean and standard deviation of Xbar?
(iii) Find P(Xbar>=2.5). (Hint:Use the fact that a sum of independent normal random variables, and hence also the sample mean, is normally distributed.)