THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT 1302 PROBABILITY AND STATISTICS II
(201011)
EXAMPLE CLASS 6
1. The Pareto distribution has been used in economics as a model for a density function
with a slowly decaying tail:
f
(
x

x
0
, θ
) =
θx
θ
0
x

θ

1
,
x
≥
x
0
,
θ >
1
.
Assume that
x
0
>
0 is known and that
X
1
, X
2
, . . . , X
n
is an i.i.d. sample.
(a) Find the methodofmoments estimator of
θ
.
(b) Find the mle of
θ
, quote the theorem for the largesample properties of mle,
and find its largesample mean and variance.
2. Let
X
1
, . . . , X
n
is an i.i.d. sample from a Rayleigh distribution with
θ >
0:
f
(
x

θ
) =
x
θ
2
e

x
2
/
2
θ
2
,
x
≥
0
.
(a) Find the methodofmoments estimator of
θ
.
(b) Find the mle of
θ
, quote the theorem for the largesample properties of mle,
and find its largesample mean and variance.
3. Specify the null and alternative hypotheses in terms of unknown parameters for the
following problems.
(a) A property developer suggests to build a private residential estate by the lake if
no more than 21,000 fish live in the lake. This project is expected to be highly
profitable. However, an environmentalist objects strongly to the project as the
living environment of the fish will be seriously damaged. Give your answer
i. if you were the property developer;
ii. if you were the environmentalist.
1
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(b) A coin has a probability
p
to turn up a head when tossed.
You want to
determine if the coin is fair.
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 Spring '11
 Dr.Yun
 Normal Distribution, Probability distribution, Probability theory, probability density function, Likelihood function, ln xi

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