# Information for second exam

Date: Friday, October 24

Place: ECJ 1.204 ( Not the usual classroom, but the same room as the second exam.)

Exam will cover: Everything covered in class and homework (reading, practice problems and written problems) through (and including) Friday, October17.

Suggestions for studying:

1. Carefully study the first exam 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. See also the suggestions for the first exam.

• 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.

I. Review questions

1. The cumulative distribution function (cdf) of a random variable X is

2. The probability mass function of a discrete  random variable X is

3. The probability density function of a continuous random variable X is

4. If X is a continuous random variable with cdf F(x), then the probability density function (pdf) of X is

5. If X is a continuous random variable with pdf f(x), then the cdf of X can be calculated by

6. If X is a discrete  random variable with probability mass function p(x), then the cdf of X can be calculated by

7. If X is a discrete  random variable with probability mass function p(x), then the expected value of X is

8. If X is a continuous random variable with pdf f(x), then the expected value of X is

9. Two other names for the expected value of X are ______________ and ______________________.

10. If you know the probability mass function p(x) of the discrete random variable X and if g(x) is a function defined on the range of X, then you can find the expected value E(g(X)) of g(X) by

11. If you know the pdf f(x) of the continuous random variable X and if g(x) is a function defined on the range of X, then you can find the expected value E(g(X)) of g(X) by

12. E(aX + b) =

13. If X is a random variable, then:

a. The definition of the variance Var(X) is Var(X)  =

b. Another formula for Var(X) which is often easier to use is

c. What Var(X)  measures intuitively is

14. If X is a random variable and c is a constant, then

a. Var (X + c) = ______________

b. Var (cX) = _________________

II. Problems from the book:
• pp. 120 - 121 #6(a) and (b) and #19 (Provide more detail than the solution in the back of the book, including breaking the first step up to explain where it comes from.)
•  p. 184 #1
•  p. 235 -237 #1, 2, 3, 4, 6, 14(a) and 14(b)

1. State Bayes Rule (Proposition 3.1) in the case of two events, calling them F and Fc rather than F1 and F2. Show how to derive this formula, starting with the definition of conditional probability and properties of union, intersection, and complements.

2. In the situation of Problem 1 on p. 184, also:
a. Find and sketch the probability mass function of X.
b. Find and sketch the cumulative distribution function of X.
c. Find Var(X) and SD(X).
d. Find E(X2)

3. In the situation of Problem 1 on p. 235, also:
a. Find and sketch the cumulative distribution function of X.
b. Find the expected value and variance of X.

4. In the situation of Problem 14 on p.237, also find the probability distribution function of X.

5. If E(X) = 2 and Var(X) = 3, find E(X2).