M 362K, Spring 03, Smith

## Assignment for Friday, November 7

*Reading*: Read the rest of Section 6.1

*To Turn In*:

1. An absented minded professor does not remember which of his 12 keys will
open his office door. He tries them at random, not remembering which ones
he has already tried.

a. On average, how many keys does he need to try before
his door opens? (Be sure you give complete reasons -- e.g., what kind of
random variable you are using, and why)

b. What is the probability that he opens his door on the
third try?

c. What is the probability that he opens his door in at
most three tries?

d. What is the probability that it takes at least three
tries to open his door?

(Calculate out all answers numerically, as well as showing your steps and
reasoning. Check to make sure your numerical answers are reasonable.)

2. Let X be a geometric random variable with parameter p. Suppose n and m
are non-negative integers.

a. Find P( X > n + m | X > m). How does your answer
compare to P(X > n)?

b. Find P(X is even). Work your answer out to a fairly
simple expression to receive full credit.

3. Both empirical and theoretical evidence suggests that a normal distribution
is a good model for heights of people of the same sex. Based on the Public
Health Service's Health and Nutrition Examination Survey in the late 70's,
the mean and standard deviations of American males aged 18 to 74 were 5 ft.
9 inches and 3 inches, respectively. The mean and standard deviation for
American Males aged 18 to 24 were 5 feet 10 inches and 2.8 inches, respectively.
Find

a. The proportion of American males aged 18 to 74 at the
time of the survey who were greater than 6 feet tall.

b. The proportion of American males aged 18 to 24 at the
time of the survey who were greater than 6 feet tall.

c. The proportion of males over six feet tall in a population
with mean 5 feet 10 inches tall and standard deviation 3 inches. Compare
with your answers to (a) and (b).

4. p. 230 #23.