10/11
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
Assignment 4
Due Date: November 17, 2010
(Hand in your solutions for Questions 4, 7, 8, 11, 14, 25, 27, 32)
1.
Two fair dice are rolled. Find the joint probability mass functions of
X
and
Y
when
(a)
X
is the largest value obtained on any die and
Y
is the sum of the values;
(b)
X
is the value on the first die and
Y
is the larger of the two values;
(c)
X
is the smallest and
Y
is the largest value obtained on the dice;
2. Consider a sequence of independent Bernoulli trials, each of which is a success with
probability
p
. Let
be the number of failures preceding the first success, and let
be the
number of failures between the first two successes. Find the joint mass function of
and
.
1
X
2
X
X
1
2
1
2
X
3.
A bin of 5 transistors is known to contain 2 that are defective. The transistors are to be tested,
one at a time, until the defective ones are identified. Denote by
the number of tests made
until the first defective is spotted and by
the number of additional tests until the second
defective is spotted; find the joint probability mass function of
and
.
1
N
N
2
N
N
4.
Consider the following joint pmf of
X
and
Y
.
Y
X
0
1
2
0
0.1
0.3
0.05
1
0.2
0.25
0.1
Find
(a)
,
;
()
Y
X
P
=
()
Y
X
P
>
(b) the marginal pmfs of
X
and
Y
;
Y
X
+
()
X
Var
(c) the pmf of
;
(d)
,
,
,
;
()
X
E
()
Y
E
()
Y
Var
(e)
,
;
()
Y
X
Cov
,
()
Y
X
Corr
,
(f) the conditional pmf of
Y
given
1
=
X
.
5. Suppose that a point is uniformly chosen on a square of area 1 having vertices (0,0), (0,1),
(1,0) and (1,1). Let
X
and
Y
be the coordinates of the point chosen.
(a) Find the marginal pdfs of
X
and
Y
.
(b) Are
X
and
Y
independent?
(c) Find the probability that the distance from (
X
,
Y
) to the center of the square is greater than
4
1
.
P. 1
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6.
The joint probability density function of
X
and
Y
is given by
()
( )
⎩
⎨
⎧
∞
<
<
≤
≤
−
−
=
−
otherwise
0
0
,
if
,
2
2
y
y
x
y
e
x
y
c
y
x
f
y
.
(a) Find the value of
c
.
(b) Find the marginal densities of
X
and
Y
.
(c) Find
.
()
X
E
7.
The joint probability density function of
X
and
Y
is given by
()
⎪
⎩
⎪
⎨
⎧
<
<
<
<
⎟
⎠
⎞
⎜
⎝
⎛
+
=
otherwise
0
2
0
,
1
0
if
2
,
2
y
x
xy
x
c
y
x
f
.
(a) Are
X
and
Y
independent?
(b) Find the value of
c
.
(c) Find the marginal probability density function of
X
.
(d) Calculate
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 Spring '08
 SMSLee
 Statistics, Probability, Probability theory, probability density function, Corr

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