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HW3 AMS 570
1. Consider a shipment of 1000 items into a factory. Suppose the factory can tolerate
about 5% defective items. Let X be the number of defective items in a sample
without replacement of size n=10. Suppose the factory returns the shipment if X ≥2.
Please obtain the probability that the factory returns a shipment of items which has
defective items.
If
items,
replacement then is a
( )
(
and sampling is done without
with pmf:
)( )
( )
If the factory returns a shipment when
none or only one item sampled is defective.
we would like the probability that either
Hence
(
)
(
[
[ (
)( )
(
)
)
(
(
)( )
(
)
)]
]
2. In a lengthy manuscript, it is discovered that only 13.5% of the pages contain no
typing errors. If we assume that the number of errors per page is a random variable
with a Poisson distribution, find the percentage of pages that have exactly one error.
Let X denote the number of errors per page. Then
X ~ Poisson( )
P( X  0) 
0
0!
 P( X  1) 
e    13.5%    2
21
e
1
2
 27.1%
3. Let X1 and X2 be independent random variables. Let X1 and Y=X1+X2 have chisquare distributions with r1 and r degrees of freedom, respectively. Here r1<r. Show
X2 has a chi-square distribution with r-r1 degrees of freedom.
1
( )
(
( )
( )
)
( )
(
)
(
)
( )
(
( )
)
(
)
(
)
4. Let X1,X2, . . .,Xn represent a random sample from a population with the pdf:
f(x; θ) = θxθ−1, 0 < x < 1, 0 < θ < ∞, zero elsewhere. Please find the mle ̂ of θ.
Answer:
n
, X n )   n ( X i ) 1
(a ) L( ; X 1 ,
i 1
n
log L  n log( )  (  1) log(X i )
i 1
 log L
1
 n *   log(X i )  0

 i 1
n
ˆ   n
 log(Xi )
n
i 1
5. Suppose
are iid with pdf (
Find the mle of (
)
( )
).
Solution. We use the mle of to estimate ( )
(
the mle of (
). The log-likelihood function is
( )
Setting
( ̂)
, zero elsewhere.
); from Theorem 6.1.3, ( ̂) is
∑
, we have
2
̂
Hence the mle of (
̂
̅
) is
̂)
(
6. Let
̂
∑
̂
̅
be a random sample from a Bernoulli distribution with parameter . If
is restricted so that we know that
, find the mle of this parameter.
Solution. The log-likelihood function here is
( )
Setting
( ̂)
(
[
)] ∑
(
)
, we have
(
̂
̂
)∑
Hence without any restrictions, the mle of
̅
̂
̂
∑
is ̅ . However, if ̅
(
maximizes the likelihood (note that, since
̅
, since
)∑
is Bernoulli, ̅
). Hence ̂
̅
3
7. Let
be a random sample from a distribution with one of two pdfs. If
then (
)
,
√
. If
, then (
)
[ (
,
)],
. Find the mle of .
Solution. Let
denote the given random sample. Recall that the mle is given by
̂
Here,
(
)
, where
(
)
∏
√
and
Hence we let ̂
8. X1 ,
if (
)
(
)
(
), and ̂
∏
[ (
)]
otherwise.
, X n iid Gamma ( p,  ) with pdf:
 p x p 1e  x
for x  0 .
( p )
Please derive the method of moments estimators of the two model parameters.
f ( x | p,  ) 
Solution:
The first two moments of the above gamma distribution are
4
E p , ( X ) 
p

, E p , ( X 2 ) 
p( p  1)
2
.
The method of moments estimator solves
pˆ
X  0
ˆ

n
X i2
pˆ ( pˆ  1)
0
n
ˆ 2
which yields
i 1

ˆ 
X

n
i 1
X i2
n
X2
and
pˆ  X ˆ  X
X

.
n
X i2
i 1
n
X2
5