Download Practice Exam 1

1234567
Math 321
Practice Exam 1 (Chapter 1, 2 in Bain and Engelhardt, 2nd)
April 5, 2063
S. K. Hyde
1234567
Name:
Raw:
(out of
Percent:
)
Show all your work to receive credit. All answers must be justified to get full credit.
True or False
Circle T or F corresponding to the best answer. (2 pts each)
1. T F If a sample space contains uncountably infinite outcomes, it is called a discrete distribution.
2. T F An event is called an elementary event if it contains exactly one outcome of the experiment.
3. T F An ordered arrangement of a set of objects is known as a combination.
4. T F The function f (x) = P [X = x] that assigns the probability to each possible value x is called the
continuous probability density function.
5. T F A random variable, say X, is a function defined over a sample space, S, that associates a real number,
X(e) = x, with each possible outcome e in S.
6. T F A distribution is symmetric if and only if µ3 = 0.
Fill in the Blank
Write the best answer in the spaces provided
7. (4 pts) The probability distribution function (pdf) for a discrete random variable must satisfy two
requirements. State both of them.
8. (3 pts) The number of distinguishable permutations of n objects of which r are of one kind and n − r are of
another kind is
9. (4 pts) State one of DeMorgan’s Laws
10. (2 pts each) If the variance of a distribution is zero, it is called a
distribution; distributions of this type have all their probability concentrated at a
value.
11. Answer the following:
(a) (6 pts) Give the general formulas for:
P (A ∩ B) =
P (A ∪ B) =
P (A | B) =
(b) (2 pts) One of the above formulas can be simplified if A and B are mutually exclusive. Give the
simplified version of the proper formula.
(c) (2 pts) One of the above formulas can be simplified if A and B are independent. Give the simplified
version of the proper formula.
Practice 1, page 2
Show Your Work
Show all work clearly and neatly. No work shown means no credit will be given. Use correct notation to get full
credit. Reserve scratch paper work for scratch paper, which means only include necessary work on the exam. Erase
all mistakes neatly. Keep it neat!
Work all the following problems on the blank COLORED sheets of paper (not white paper or scratch paper)
provided by me. If you need more, get some more from the testing center personnel. I am providing the paper.
Hence, I will require you to do the following:
(a) Start each numbered problem on a NEW sheet of paper (Do not start a problem on the back side of a paper!).
You do not need to start a different lettered problem on a new sheet. (5 pts)
(b) Make sure the problem numbers are clearly labeled (even after pages are stapled together (e.g. not covered
by a staple)) Ask to staple your test yourself to make sure. (5 pts)
(c) Make sure the problems are in correct numerical/alphabetical order. (5 pts)
12. (10 pts) Show that
n
n!
n − r1
n − r1 − r2 − · · · − rk−1
,
···
=
r1 !r2 ! · · · rk !
r1
r2
rk
where n =
Pk
i=1 rk .
13. (10 pts) Prove: If X is a random variable, then E (X − µ)2 = E(X 2 ) − µ2
14. (10 pts) Find the moment generating function for the pdf
f (x) = pq x−1 , x = 1, 2, · · ·
15. Assume that X is a continuous random variable with pdf

−(x+2)

, −2 < x < ∞
e
f (x) =


0,
otherwise
(a) (10 pts) Find the moment generating funciton of X.
(b) (10 pts) Use the MGF of 15a to find E(X).
16. A bag contains five blue balls and three red balls. A boy draws a ball, and then draws another without
replacement. Compute the following probabilites:
(a) (5 pts) P (2 blue balls)
(b) (5 pts) P (1 blue and 1 red)
(c) (5 pts) P (at least 1 blue)
(d) (5 pts) P (2 red balls)
17. (28 pts) The function IA (x) is called the indicator function. It “indicates” where the function is non-zero. In
other words,
(
1, if x ∈ A
IA (x) =
0, if x ∈
/A
Let c be a constant and consider the pdf
f (x) = cx2 I[0,1] (x).
(a) Find c.
(b) F (x)
(c) E(X r )
(d) µ
(e) σ
(f) P (x > .5)
Practice 1, page 3
(g) P (x 6 .3)
18. (10 pts) Prove: If A is an event and A′ is its complement, then
P (A) = 1 − P (A′ )
19. (10 pts) The expected value of a function of a random variable can be used to place bounds on the
probability of a random variable. For example, if X is a random variable and u(x) is a nonnegative
real-valued function, then for any positive constant c,
P [u(X) > c] 6
E(u(X))
.
c
Using this inequality, prove the following form of Chebychev’s inquality:
P [|X − µ| > kσ] 6
1
.
k2
20. (10 pts) The cumulant generating function, ψ(t), is defined as the natural logarithm of the moment
generating function. In other words,
ψ(t) = ln MX (t).
The value of the rth derivative of ψ(t) evaluated at t = 0 is κr = ψ (r) (0). Show that κ2 = σ 2 , where σ 2 is the
variance of X.
21. (extra credit) (5 pts) Compute the probability of drawing a two pair hand in poker.
22. (extra credit) (5 pts) A pair of events A and B cannot be simultaneously mutually exclusive and independent.
Suppose P (A) > 0 and P (B) > 0. Prove that if A and B are mutually exclusive they cannot be independent.