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:

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

Exam will cover:

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.

Exam format:

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.

Please Note:

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 f_X,Y(x,y), then for a function g(x,y),
E(g(X,Y)) =_________________.

3. If X and Y are random variables with joint pmf p_X,Y(x,y), then for a function g(x,y),
E(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 f_X and f_Y , then the pdf of their sum can be calculated by f_X+Y(u) = ____________________________________.

10. If X and Y are independent discrete random variables with pmf's p_X and p_Y , then the pmf of their sum can be calculated by p_X+Y(u) = ____________________________________.

11. If X and Y are independent normal random variables with means µ_X and µ_Y, respectively, and standard deviation sigma_X and sigma_Y, respectively, then their sum is _______________________ with parameters ______ and ____________.

12. If X and Y are independent random variables, then E(XY) = _____________.

13. If X₁, X₂, ... , X_n are independent, identically distributed random variables with mean µ and standard deviation sigma, and if n is large enough, then their sum X₁+ X₂+ ... + X_n 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

II. More:

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 is false).
    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 F_Y(y) of Y in terms of the cumulative distribution function F_X(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 f_Y(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 F_Y(y) of Y in terms of the cdf F_X(x) of X.
    b. Find a formula for the pdf f_Y(y) of Y in terms of the pdf f_X(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.