Stat 116 Homework 8
Due Wednesday, June 4th.
Please show work and justify answers. No credit for a final answer with no explanation,
even if the answer is correct.
1. Prove that Binomial random variables
Bin
(
n, p
) converge in distribution to a poisson
random variable with parameter
λ
as
n
tends to infinity, if
np
=
λ
(with
λ
fixed).
Solution:
The MGF of a
Bin
(
n, p
) is (
pe
t
+ 1

p
)
n
. Now let
n
→ ∞
with
np
=
λ
,
fixed. Then the MGF become
(
pe
t
+ 1

p
)
n
=
λ
n
e
t
+ 1

λ
n
n
=
1 + (
e
t

1)
λ
n
n
→
e
λ
(
e
t

1)
as
n
→ ∞
since (1 +
x/n
)
n
→
e
x
and so we recognize the above quantity as the MGF
of a
Poisson
(
λ
) r.v. .
2. Let
X
i
, i
= 1
,
2
, ..., n
be independent Bernoulli(
p
) random variables and let
Y
n
=
1
n
∑
n
i
=1
X
i
.
Recall, that if
U
n
converges in distribution to
W
and
V
n
converges in prob
ability to
c
then by Slutsky’s Theorem
U
n
V
n
converges in distribution to
cW
, you may
use this result for this problem.
(a) Show that
√
n
(
Y
n

p
)
→
N
[0
, p
(1

p
)].
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '07
 Staff
 Binomial, Probability, Probability theory, Trigraph, yn

Click to edit the document details