SOLUTION FOR HOMEWORK 4, STAT 4382
Well, welcome to your fourth homework. Exam 1 is based on all 4 homeworks.
1. Problem 2.1.4. Here
X
is
Binomial
(
p,N
) given
N
which is a
Binomial
(
q,m
) random
variable. We are asked to find the mean of
X
. Write,
E
(
X
) =
E
(
E
(
X

N
)) =
E
(
pN
) =
pE
(
N
) =
pqm.
I skip plugging numbers.
2. Problem 2.1.9. Here
N
is
Poisson
(
λ
), and given
N
the random variable
X
is discrete
uniform on the set
{
0
,
1
,...,N
+ 1
}
. We are asked to find the marginal distribution of
X
.
Well, we begin with the joint distribution
p
X,N
(
x,n
) =
p
N
(
n
)
p
X

N
(
x

n
) = [
e

λ
λ
n
/n
!][1
/
(
n
+ 2)]
I
(
x
∈ {
0
,
1
,...,n
+ 1
}
)
.
Now we can find the marginal (pay attention to the fact that given
x
the random
N
cannot
be smaller than
x

1
p
X
(
x
) =
∞
summationdisplay
n
=min(0
,x

1)
e

λ
λ
n
/
[
n
!(
n
+ 2)]
=
∞
summationdisplay
n
=min(0
,x

1)
e

λ
λ
n
[
1
(
n
+ 1)!

1
(
n
+ 2)!
]
.
I stop here and will allow you to finish. Note that we arrived at difference of two terms
where each term can be calculated using the fact that a Poisson probability mass function
is added to 1.
3. Problem 2.3.2. Here
N
is
Binomial
(
p,m
) with
p
= 1
/
2 (fair coin) and
m
= 6. Then,
given
N
,
Z
is
Binomial
(
q,N
) with
q
= 1
/
2. First, we are asked to find the mean and
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 Fall '11
 krwaw
 Binomial

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