Stat 5101 (Geyer) Fall 2011
Homework Assignment 12
Due Wednesday, December 14, 2011
Solve each problem. Explain your reasoning. No credit for answers with
no explanation. If the problem is a proof, then you need words as well as
formulas. Explain why your formulas follow one from another.
121.
Give the details of the argument that the Poi(
μ
) distribution is ap
proximately normal when
μ
is large.
122.
Suppose
X
1
,
X
2
,
...
are IID with mean
μ
and variance
σ
2
and
X
n
=
1
n
n
X
i
=1
X
i
What is the approximate normal distribution of sin(
X
n
) when
n
is large?
123.
Suppose
X
1
,
X
2
,
...
are IID Poi(
μ
) random variables and
X
n
=
1
n
n
X
i
=1
X
i
To what random variable does
√
n
(
e

X
n

e

μ
)
converge in distribution?
124.
Suppose
X
1
,
X
2
,
...
are IID Ber(
p
) random variables with 0
< p <
1
and
X
n
=
1
n
n
X
i
=1
X
i
(a) What is the approximate normal distribution of
X
n
(1

X
n
) when
n
is
large?
(b) There is something unusual about the case
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 Spring '11
 Staff
 Standard Deviation, Variance, Probability theory, probability density function, yn, approximate normal distribution

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