M362K, Smith, Fall 03
Information for Third Exam
Date: Friday, November 21
Place: ECJ 1.204 ( Not the usual classroom, but the same
room as the previous exams.)
Exam will cover: Everything covered in class and homework (reading,
practice problems and written problems) through (and including) Friday,
November 14. This includes: Bernoulli, binomial, Poisson, geometric, normal,
and exponential random variables (what they are; when they are used; their
pdf/pmf's, expectations, and variances; which ones are related and how; using
them); finding the cmf and pmf of a function of a random variable; joint
distributions (joint pmf's, pdf's, cmf's; marginal distributions; using joint
pmf's and pdf's to find marginal distributions; using pdf's and pmf's to
find probabilities), independent random variables.
Normal tables: You will be provided with a copy of the table on p.
203 giving probabilities for the normal distribution.
Suggestions for studying:
1. Carefully study the first two exams and returned homework to understand
things you may have done wrong, including inadequate explanation, improper
use of notation, etc. Consult the solutions on reserve in the PMA library
as needed.
2. Here are some suggested practice questions and problems.
Please Note:
 To get the most out of the selftest problems in the book, give them
a serious try before looking at the solutions in the back of the book.
 Remember that some of the "solutions" in the book include less detail
(especially reasons) than I expect of you.
 These problems are intended to give you additional practice with the
concepts, vocabulary, notation, etc., and with problem solving involving
this material. Do not assume that exam problems will be just like
these (although some might be!).
I. Review Questions
1. Complete the following chart:
Random Variable

Definition and/or When It Is
Used

Parameters

pdf/pmf (formula
and graph)

E(X)

VAR(X)

Connections with other random variables

Bernoulli







Binomial







Geometric







Poisson







Exponential







Normal







2.If X and Y are discrete random variables, then their joint pmf is defined
by:
3.If X and Y are random variables, then the joint cdf of X and Y is defined
by:
4. If X and Y are continuous random variables, then their joint cdf is:
5. If X and Y are continuous random variables with joint pdf f, then their
joint cdf F can be calculated by:
6. If X and Y are discrete random variable with pmf p, then their joint
cdf can be calculated by:
7. If X and Y are jointly distributed, then the distributions of X and Y
alone are called _________________ distributions.
8. If X and Y are discrete random variables with joint pmf f(x,y), then
the maraginal pmf's of X and Y can be calculated by:
9. If X and Y are continuous random variables with joint pdf f(x,y), then
the marginal pdf’s of X and Y can be found by:
10. Random variables X and Y are said to be independent if:
II. Review Problems from Textbook
 p. 185 #9, 13 (What kind of random variable models this situation?
Why?), 15(a), 18
 p. 236 #9, 10, 11, 12, 13
 p. 299#1a, b; 3a, b, c; 5, 6
III. Additional Review Problems
1. In each of the following situations, first decide what type of random
variable(s) is (are) involved, and explain why. (In particular, state any
assumptions you are making.) Then use what you know about the random variable(s)
involved to solve the problem.
A. From an ordinary deck of 52 cards, we draw cards at random, with replacement,
until a seven is drawn.
a. What is the probability that exactly 5 draws are needed?
b. What is the probability that at least 5 draws are needed?
B. From an ordinary deck of 52 cards, we draw 5 cards at random, with replacement.
What is the probability that we get two sevens?
C. From an ordinaty deck of 52 cards, we draw a card. You get a dollar if
it is a seven, nothing otherwise. What is the probability that you get a
dollar?
D. On a certain summer evening, shooting stars are observed at a
rate of one every 12 minutes. What is the probability that three shooting
stars are observed in 30 minutes?
E. A restaurant serves three seafood entrees, four red meat entrees,
two poultry entrees, and one vegetarian entree. Suppose customers select from
these randomly.
a. What is the probability that exactly two of the next
five customers order seafood entrees?
b. What is the probability that none of the next four customers
orders a seafood entree, but the fifth does?
F. Suppose that 90% of the patients with a certain disease can be cured
with a certain drug. What is the approximate probability that of 50 such patients,
at least 45 can be cured with the drug?
2. The joint pmf of the random variables X and Y is given in the following
table.

Y

X


1

2

3

4

0

0

1/8

1/8

1/8

1

0

0

1/8

1/8

2

1/8

1/8

0

1/8

a. Find the marginal pmf of Y.
b. Find the value of the joint cdf of X and Y at (2,2).
3. X is an exponential random variable with parameter 3. Find the pdf of
each of the following random variables:
a. U = 2X. (What kind of random variable is U? You should
be able to tell when you get the pdf.)
b. Y = X^{2}. (Yes, your answer will be messier
than your answer to part (a).)
4. Explain (using pictures, equations, and words as needed to explain your
reasoning clearly) how you can find each of the following quantities from
the table on p. 203. (Assume Z is a standard normal random variable.)
a. P{Z ≤ .81} b.
P{Z ≥ .81} c. P{Z ≤ .81} d. P{Z ≥ .81}
e. P{.81 ≤ Z ≤ .81}
f. C with P{Z ≤ C} = .81 g. C with P{Z
≥ C} = .81 h. C with P{C ≤ Z ≤ C} = .81
5. You have heard that plain m&m’s contain 30% brown candies. However,
you are a peanut m&m’s aficionado, and your extensive experience with
peanut m&m’s suggests that they contain fewer than 30% brown, an unfortunate
situation, since your prefer brown  they seem more chocolatey. Deciding
to gather some hard evidence and apply your knowledge of probability to the
matter, you carefully keep track of the next 300 peanut m&m’s you consume.
You discover that 78 of them are brown. Approximately what is the probability
of this happening if the percent of peanut m&m’s which are brown is indeed
30%? Approximately what is the probability of obtaining less than 78 brown
m&m's (still assuming the percent of brown is indeed 30%)? Does your
answer provide support for your suspicion or not?