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 X_{1},
X_{2} that have the same probability function in problem 1
above. Let Y=X_{1}+X_{2} . 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 X_{1} and X_{2} are
independent, P(Y=y)= Sum_{a=1}^{5} P(X_{1}=a and
X_{2}=y-a) =
Sum_{a=1}^{a=y-1} P(X_{1}=a) P(X_{2}=y-a) =
Sum_{a=1}^{a=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 X_{1}, X_{2}, X_{3}, ...,
X_{16} 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.)

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