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.

- 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)

III. Additional problems:

1. State Bayes Rule (Proposition 3.1) in the case of two events, calling them F and F

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(X

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(X