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.
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.
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 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).