Stat 421: Probability Theory
Mid Term I
Fall 2012.
Name
Show all your works to receive full credit.
1. This problem is for undergraduate students ONLY.
(a) Prove that for every two events A and B , the probability that exactly one of the
two events will o
Stat 421 Solutions for Homework Set 5
Page 129 Exercise 5: Suppose that the joint p.d.f. of two random variables X and Y is as follows:
cfw_
c(x2 + y ) for 0 y 1 x2 , 1 < x < 1
f (x, y ) =
0
otherwise.
Determine (a) the value of the constant c; (b) P r(0
Stat 421 Solutions for Homework Set 6
Page 151 Exercise 3: Suppose that a point (X, Y ) is chosen at random from the disk S dened as follows:
S = cfw_(x, y ) : (x 1)2 + (y + 2)2 9.
Determine (a) the conditional p.d.f. of Y for every given value of X , and
Stat 421: Probability Theory
Mid Term II
Fall 2012.
Do THREE problems in class and take FOUR home. Show all your works to receive full
credit.
1. (U) Let X1 and X2 have a joint pdf
cfw_
fX1 ,X2 (x1 , x2 ) =
8x1 x2 0 < x1 < x2 < 1
.
0
Otherwise
(a) Find th
Stat 421 Solutions for Homework Set 9
Page 240 Exercise 10: Suppose that the random variables X and Y are iid and that the mgf of each is:
2
M (t) = et +3t for < t < . Find the mgf of Z = 2X 3Y + 4.
2
MX (t) = et +3t , < t <
. So:
2
MY (t) = et +3t , < t
Stat 421 Solutions for Homework Set 8
Page 225 Exercise 6: Suppose that a particle starts at the origin of the real line and moves along the line
in jumps of one unit. For each jump the probability is p (0 p 1) that the particle will jump one unit to
the
Stat 421 Solutions for Homework Set 7
Page 174 Exercise 3: Suppose that the p.d.f. of a random variable X is as follows:
cfw_
1
x for 0 x 2
f (x) =
2
0
otherwise.
Also, suppose that Y = X (2 X ). Determine the c.d.f. and the p.d.f. of Y .
Solution: As x v
Stat 421 Solutions for Homework Set 4
Page 100 Exercise 5: Suppose that a box contains seven red balls and three blue balls. If ve balls are selected at random, without replacement, determine the p.m.f. of the number of red balls that will be obtained.
So
Stat 421 Solutions for Homework Set 3
Page 65 Exercise 5: A box contains r red balls and b blue balls. One ball is selected at random and its
color is observed. The ball is then returned to the box and k additional balls of the same color are also put
int
Stat 421 Solutions for Homework Set 2 Page 32 Exercise 8: An elevator in a building starts with five passengers and stops at seven floors. If every passenger is equally likely to get off at each floor and all the passengers leave independently of each oth
Stat 421 Solutions for Homework Set 1
Page 15 Exercise 1: Suppose that A B . Show that B c Ac .
Proof. Assume that x B c . We need to show that x Ac . Let us show this indirectly. By way of
contradiction we assume, that x A. Then x B because A B . (contra
Stat 421: Probability Theory
Fall 2012
Solutions to Mini Homework Set 1
1. Show that
(a) (A B )c = Ac B c and
(b) (A B )c = Ac B c .
/
/
/
Proof. (a) Observe that, x (A B )c x (A B ) x A and x B x Ac and
c
c
c
c
c
c
c
x B x (A B ). Therefore, (A B ) (A B