36226 Summer 2010
Homework 4
Due July 12
1. Assume that
X
, the proportion of defective products that a machine produces in a day, has
the following density:
f
X
(
x

θ
) =
θ
(1

x
)
θ

1
I
(0
,
1)
(
x
)
.
Note that this is a Beta(1,
θ
) distribution.
In order to estimate
θ
, the machine was observed for 100 days and the proportion of defective
products was recorded for each day. The data is in the
R
file
Defects.Rdata
that is linked
from the HW section of the course website.
The name of the dataset is
defects
.
It is a
vector with 100 entries.
(a) Find a method of moments
estimator
for
θ
based on a random sample of size
n
from the
above distribution.
(b) Find a maximum likelihood
estimator
for
θ
.
(c) Use part (a) and the data to find an
estimate
for
θ
.
(d) Use part (b) and the data to find a second
estimate
for
θ
.
(e) Use your estimates from part (c) and (d) to estimate the probability that on a given
day, the proportion of defective products will exceed .025
2. Let
X
1
, X
2
, . . . , X
n
be a random sample from the following distribution:
f
X
(
x

θ
) =
3
x
2
θ
3
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 Summer '09
 Normal Distribution, Maximum likelihood, Estimation theory, Bias of an estimator

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