09/10
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
Assignment 2
Due Date: October 9, 2009
(Hand in your solutions for Questions 4, 7, 13, 23, 24, 31, 34, 36)
1.
Two balls are chosen randomly from an urn containing 6 white, 3 black, and 1 orange balls.
Suppose that we win $1 for each white ball drawn and we lose $1 for each orange ball drawn.
Denote $
X
as the amount that we can win.
(a) What are the possible values of
X
?
(b) Determine the probability mass function and cumulative distribution function of
X
.
2.
Two fair dice are rolled. Let
X
be the product of the 2 dice. Determine the pmf of
X
.
3. A jar contains
chips, numbered 1, 2, …
n
m
+
n
m
+
. A set of size
n
is drawn. If we let
X
denote the number of chips drawn having numbers that exceed all of the numbers of those
remaining, determine the probability mass function of
X
.
4.
For each of the following, determine the constant
c
such that
( )
x
p
satisfies the conditions of
being a pmf for a random variable
X
, and then compute the mean and variance.
(a)
,
;
()
cx
x
p
=
10
,...,
3
,
2
,
1
=
x
(b)
()
x
c
x
p
=
,
;
4
,
3
,
2
,
1
=
x
(c)
()
()
2
1
+
=
x
c
x
p
,
;
1
,
0
,
1
−
=
x
(d)
,
.
() ( )
!
x
c
x
p
=
3
,
2
,
1
=
x
5. Let
X
be a random variable with pmf
()
( )
9
1
2
+
=
x
x
p
,
1
,
0
,
1
−
=
x
Compute
,
()
X
E
( )
2
X
E
and
( )
4
3
2
2
+
−
X
X
E
.
6.
A game uses two fair dice. To participate, you pay $20 per roll. You win $10 if even shows,
$42 if 7 shows, and $102 if 11 shows. The game is fair if your expected gain is $0. Is the
game fair? What is the variance of your gain?
7.
A box contains 5 red and 5 blue marbles. Two marbles are withdrawn randomly. If they are
the same color, then you win $1.10; if they are different colors, then you win -$1.00 (that is,
you lose $1.00). Calculate
(a) the expected value of the amount you win;
(b) the variance of the amount you win.
P. 1