STAT 410
Spring 2008
Homework #4
(due Friday, February 15, by 4:00 p.m.)
1.
Let X and Y be independent random variables, each geometrically distributed
with the probability of “success”
p
, 0 <
p
< 1. That is,
p
X
(
k
) =
p
Y
(
k
) =
( )
1
1


⋅
k
p
p
,
k
= 1, 2, 3, … ,
a)
Find P
(
X
> Y
). [ Hint: First, find P
(
X = Y
).
]
b)
Find P
(
X
+ Y =
n
),
n
= 2, 3, 4, …
, and
P
(
X =
k

X
+ Y =
n
),
k
= 1, 2, 3, … ,
n
– 1.
From the textbook:
2.3.3
Let
f
(
x
1
,
x
2
) =
3
2
2
1
21
x
x
, 0 <
x
1
<
x
2
< 1, zero elsewhere, be the joint
pdf of X
1
and X
2
.
(a) Find the conditional mean and variance of X
1
, given X
2
=
x
2
, 0 <
x
2
< 1.
(b) Find the distribution of Y = E
(
X
1

X
2
).
(c) Determine E
(
Y
) and Var
(
Y
) and compare these to E
(
X
1
) and Var
(
X
1
),
respectively.
2.3.10
Let X
1
and X
2
have joint
pmf
p
(
x
1
,
x
2
) described as follows:
(
x
1
,
x
2
)
(
0,
0
)
(
0,
1
)
(
1,
0
)
(
1,
1
)
(
2,
0
)
(
2,
1
)
p
(
x
1
,
x
2