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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
²
n
=
±
1 + (
e
t

1)
λ
n
²
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
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This note was uploaded on 01/13/2010 for the course STATS 116 taught by Professor Staff during the Spring '07 term at Stanford.
 Spring '07
 Staff
 Binomial, Probability

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