M362K, Smith, Fall 03
Information for Final Exam
Date: Saturday, December 13, 9 a.m - 12 noon
Place: CPE 2.210 ( Not the usual classroom,
and not the same room as the previous exams -- across Dean Keeton.)
Exam Week Office Hours:
Exam will cover:
- Monday, Dec. 8: 2 - 3 pm
- Wednesday, Dec. 10: 3 - 4 pm
- Thursday, Dec. 11: 4 - 5 pm
- Friday, Dec. 12: 12:30 - 1:30 pm
- Everything covered on previous exams, plus topics covered since
then: sums of independent random varaibales (including Bernoulli and normal),
conditional distributions, expected value of a function of two random variables,
expected value of a sum of random variables, covariance, variance of
a sum of random variables, central limit theorem.
- Topics covered since the last exam will account for approximately
one-third of the points on the final exam, to make total exam coverage
even throughout the semester.
- Questions on the exam will be of the same general nature
and level of difficulty as the questions on previous exams. (Do not expect
them to be just the same with only minor changes, however.)
- The exam will be about twice as long as a midsemester exam, but
you will have three hours to work on it.
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 midsemester 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.
- To get the most out of the self-test 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 ON MATERIAL SINCE THE LAST EXAM
1. E(aX + bY) = ___________________
2. If X and Y are random variables with joint pdf fX,Y(x,y),
then for a function g(x,y),
3. If X and Y are random variables with joint pmf pX,Y(x,y),
then for a function g(x,y),
4. Var (X + Y) = Var (X) + Var(Y) provided ______________________________.
5. If X and Y are random variables, then:
a. The definition of the covariance of X and Y is
Cov(X,Y) = ______________________________.
b. Another formula for Cov(X,Y) which is often easier to use is
Cov (X,Y) = ______________________________.
c. Intuitively, Cov(X,Y) measures ___________________________.
d. If X and Y are independent, then Cov(X,Y) = __________.
e. With no assumptions about X and Y, Var (X + Y) = __________________________________.
6. For discrete random variables X and Y the conditional probability density
function of X given Y is _________________________. (Give your answer in
two forms -- one involving the pmf and one not using the pmf.)
7. For discrete random variables X and Y the conditional probability density
function of X given Y is _________________________.
8. The sum of n independent identically distributed Bernoulli random variables
with parameter p is a ___________________ random variable with parameters
____ and ______.
9. If X and Y are independent continuous random variables with pdf's fX
and fY , then the pdf of their sum can be calculated by fX+Y(u)
10. If X and Y are independent discrete random variables with pmf's pX
and pY , then the pmf of their sum can be calculated by
pX+Y(u) = ____________________________________.
11. If X and Y are independent normal random variables with means µX
and µY, respectively, and standard deviation sigmaX
and sigmaY, respectively, then their sum is _______________________
with parameters ______ and ____________.
12. If X and Y are independent random variables, then E(XY) = _____________.
13. If X1, X2, ... , Xn are independent,
identically distributed random variables with mean µ and standard
deviation sigma, and if n is large enough, then their sum X1+
X2+ ... + Xn is approximately _______________ with
mean ____________ and standard deviation _____________.
14. If you forget what the mean and variance of a binomial random variable
are, how can you combine some things we've studied recently to easily figure
out what they are?
REVIEW PROBLEMS ON MATERIAL SINCE THE LAST EXAM
I. From the textbook:
- p. 301, # 12
- p. 383, #33
- p. 392, #19
- p. 428, #13a and b, 14
- p. 431, #7, 8, 9
1. X and Y are independent exponential random variables, each with the
same parameter lambda. Let U be the minimum of X and Y.
a. What is the joint pdf of X and Y?Why?
b. Find the cdf of U. [Hint: U ≤ u means either X ≤
u or Y ≤ u or both.]
c. Find the pdf of U.
2. In ethnic group A, 60% of voters prefer candidate C to candidate
D. In ethnic group B, 40% of voters prefer C to D. You take a poll of 200
voters from group A and an independent poll of 300 voters from group D.
Let P be the proportion of voters in your poll of group A who prefer
C. Let Q be the proportion of voters in your poll of group B. Then
P and Q are random variables. What is the standard deviation of P - Q, the
difference in the proportions in the two polls?
3. Define Covariance. Use your definition to prove that
Cov(aX +bY , Z) = aCov(X , Z) + bCov(Y , Z)
4. Var(X) = 1, Var(Y) = 2, and Var(X + Y) = 3. Find Cov (X, Y).
5. Var(X) = 1, Var(Y) = 2, and Cov(X, Y) = 3. Find
a. Var(X + Y)
b. Var(X - Y)
6. True or false. If the statement is (always) true, prove it mathematically.
If it is false, give a counter example (that is, an example where the statment
a. Var (X + Y) = Var(X) + Var(Y)
b. If X and Y are independent, then Cov(X, Y) = 0.
c. If Cov(X,Y) = 0, then X and Y are independent.
7. For each step in the proof of Proposition 3.1 (p. 328), give the reason
why the step is valid. If necessary, insert any intermediate steps needed
so that each step has a single reason.
8. X is a certain random variable and Y = X3.
a. Find the cumulative distribution function FY(y)
of Y in terms of the cumulative distribution function FX(x)
of X. (Be sure to give its value for all possible values of y.)
b. Now suppose in addition that X is exponential with
parameter 2. Find the probabiity density function fY(y) of Y.
(Be sure to give its value for all possible values of y.)
9. X is a random variable and Y = cX
a. Find a formula for the cdf FY(y) of Y in
terms of the cdf FX(x) of X.
b. Find a formula for the pdf fY(y) of Y in
terms of the pdf fX(x) of X. [Use part (a)]
c. Now suppose X is normal with mean µ and standard
deviation sigma. Use part (b) to get a formula for the pdf of Y.
d. Use algebra to put your answer to part (c) in a form
that shows that Y is also normal.