Reserve Problems Chapter 7 Section 3 Problem 10
An exponential distribution is known to have a mean of 10. You want to find the standard error
of the median of this distribution if a random sample of size 8 is drawn. In order to use the
bootstrap method
5
B
n=
bootstrap samples are generated.
Sample
1
29.86
7.71
7.57
5.32
7.40
10.51
11.26
12.85
2
0.50
20.91
10.61
13.02
1.81
14.62
43.43
12.92
3
3.60
36.86
11.26
2.34
10.18
10.51
1.33
0.58
4
0.58
3.97
18.10
8.57
11.63
8.57
5.70
4.47
5
10.99
4.16
0.85
0.50
7.74
18.10
3.95
1.02
Find the standard error of the median with the bootstrap method.
SOLUTION
Sample
Sample median
1
9.11
2
12.97
3
6.89
4
7.14
5
4.06
Reserve Problems Chapter 7 Section 4 Problem 1
Consider the probability density function
7
()
67
7 , 0
()
0, 0
y
y e y
fy
y
−
−
=
1
7.35
2
5.14
3
4.74
4
4.65
5
6.18
Find the maximum likelihood estimator for
0
based on a random sample of size n. Use
variable “i” as sum index in your answer.
SOLUTION
Reserve Problems Chapter 7 Section 4 Problem 2
Consider the Pareto distribution
1,,
( ) {
0, .
x
fx x
x
+
=
Find the maximum likelihood estimator for
0
if
0
is known. Use variable “i” as sum
index in your answer.
SOLUTION
Reserve Problems Chapter 7 Section 4 Problem 3
A random sample of size n is selected from an exponential distribution
, 0,
( ) { 0, 0.
x
ex
fx x
−
=
Parameter
can take values of 1, 2, and 3.
Find the maximum likelihood estimator of
.
SOLUTION
Reserve Problems Chapter 7 Section 4 Problem 4
Consider the Pareto distribution
1,,
( ) {
0, .
x
fx x
x
+
=
(a) Find the moment estimator for
1
if
0
is known. Use M as the symbol of sample
mean.
(b) Find the moment estimator for
0
if
1
is known. Use M as the symbol of sample
mean.
SOLUTION
(a)
Reserve Problems Chapter 7 Section 4 Problem 5
The time in minutes between customer arrivals at a store has an exponential distribution with
parameter
. The prior distribution for
is exponential with the mean of 2. A random sample
of
5n=
gives the average time between arrivals
3.9x=
minutes. Find the Bayes estimate for
.
SOLUTION
Reserve Problems Chapter 7 Section 4 Problem 6
The length of the manufactured component is a normal random variable with an unknown mean
and the known variance of
2
0.3cm
. The prior distribution for µ is normal with
020cm
=
and
22
00.01cm
=
. A random sample of 10 components was taken. Their lengths in centimeters are
20.05, 19.93, 20.00, 19.93, 20.06, 20.00, 20.05, 20.04, 20.10, 19.99.
(a) Find the Bayes estimate of
.
(b) Find the probability that is greater than 20.
SOLUTION
(a)
(b)
Reserve Problems Chapter 7 Section 4 Problem 7
A random sample
1,…, n
XX
is selected from the shifted exponential distribution
,
( ) { 0,
x
ex
fx x
−
=
Is
1X−
an unbiased estimator of
?
SOLUTION
1X−
Reserve Problems Chapter 7 Section 4 Problem 8
Let X be a random variable with the following probability distribution:
( ) ( )
1 ,0 1
{0,
xx
fx otherwise
+
=
Find the maximum likelihood estimator of
based on a random sample of size n.
SOLUTION
Reserve Problems Chapter 7 Section 4 Problem 9
Consider the shifted exponential distribution
( )
( )
,
x
f x e x
−−
=
.
When
0
=
, this density reduces to the usual exponential distribution. When
0
, there is
positive probability only to the right of
.
Find the maximum likelihood estimator of
based on a random sample of size n.
Find the maximum likelihood estimator of
based on a random sample of size n.
SOLUTION
Reserve Problems Chapter 7 Section 4 Problem 10
The Rayleigh distribution has probability density function
( )
2/2x
x
f x e
−
=
,
0x
,
0
.
(a) It can be shown that
( )
22EX
=
. Use this information to construct an unbiased estimator for
.
(b) Find the maximum likelihood estimator of
.
(c) Select the right expression for the median of the Rayleigh distribution.
(d) Can we estimate the median a by substituting our estimate for
into the equation for a?
SOLUTION
(a)
Reserve Problems Chapter 7 Section 4 Problem 11
Suppose that X is a normal random variable with unknown mean
and known variance
2
.
The prior distribution for
is a normal distribution with mean
0
and variance
2
0
. Does the
Bayes estimator for
become the maximum likelihood estimator when the sample size n is
large?
SOLUTION
Reserve Problems Chapter 7 Section 4 Problem 12
Suppose that X is a normal random variable with unknown mean
and known variance
2
.
The prior distribution for
is a uniform distribution defined over the interval
,ab
.
(a) Find the posterior distribution for
.
(b) Find the Bayes estimator for
.
SOLUTION
(a)
Reserve Problems Chapter 7 Section 4 Problem 13
Suppose that X is a Poisson random variable with parameter
. Let the prior distribution for
be a gamma distribution with parameters
1rm=+
and
( )
0
1/m
=+
.
Is the posterior distribution for parameter
of random variable X a gamma distribution?
Find the Bayes estimator for parameter
of the random variable X.
SOLUTION
( )
!
x
e
fx x
−
=
for
0,1, 2,…x=
,
Reserve Supplemental Exercises Chapter 7 Problem 1
A norm population has a known mean 50 and known variance
22
=
. A random sample of
16n=
is selected from this population, and the sample mean is
54x=
.
How unusual is this result?
SOLUTION
Reserve Supplemental Exercises Chapter 7 Problem 2
Let
( )
1
f x x
−
=
,
0
, and
01x
. Find the maximum likelihood estimator for
.
SOLUTION
Reserve Supplemental Exercises Chapter 7 Problem 3
You plan to use a rod to lay out a square, each side of which is the length of the rod. The length
of the rod is
, which is unknown. You are interested in estimating the area of the square, which
is
2
. Because
is unknown, you measure it n times, obtaining observations
1 2 3
, ,…,X X X
.
Suppose that each measurement is unbiased for
with variance
2
.
(a) Is
2
X
a biased or an unbiased estimate of the area of the square?
(b) Is
22
/X S n−
a biased or an unbiased estimate of the area of the square?
SOLUTION
(a)
Reserve Supplemental Exercises Chapter 7 Problem 4
When the sample standard deviation is based on a random sample of size n from a normal
population, it can be shown that S is a biased estimator for
. Specifically,
( ) ( ) ( )
( )
2 / 1 / 2
1 / 2
nn
ES n
−
=
−
.
(a) Use this result to obtain an unbiased estimator for
of the form
n
cS
, when the constant
n
c
depends on the sample size n.
(b) Find the value of
n
c
for
10n=
and
25n=
.
SOLUTION
(a)
Reserve Supplemental Exercises Chapter 7 Problem 5
Let
12
, ,…, n
X X X
be a random sample of size n from X, a random variable having cumulative
distribution function
( )
Fx
. Rank the elements in order of increasing numerical magnitude,
resulting in
( ) ( ) ( )
12
, ,…, n
X X X
, where
( )
1
X
is the smallest sample element (
( )
12
1min , ,…, n
X X X X=
) and
( )
n
X
is the largest sample element (
( )
12
max , ,…, n
n
X X X X=
).
( )
i
X
is called the ith order statistic. Often the distribution of some of the order statistics is of
interest, particularly the minimum and maximum sample values
( )
1
X
and
( )
n
X
, respectively.
(a) Find the cumulative distribution functions of these two order statistics, denoted respectively
by
( )
( )
1
X
Ft
and
( )
( )
n
X
Ft
.
(b) If X is continuous with probability density function
( )
fx
, find the probability distributions of
( )
1
X
and
( )
n
X
.
(c) Let
12
, ,…, n
X X X
be a random sample of a Bernoulli random variable with parameter P. Find
the following probabilities.
(d) Let
12
, ,…, n
X X X
be a random sample of a normal random variable with mean
and
variance
2
. Derive the probability density functions of
( )
1
X
and
( )
n
X
.
(e) Let
12
, ,…, n
X X X
be a random sample of a exponential random variable of parameter
.
Derive the cumulative distribution functions and probability density functions for
( )
1
X
and
( )
n
X
.
SOLUTION
(a)
(b)
Reserve Supplemental Exercises Chapter 7 Problem 6
When the population has a normal distribution, the estimator
( )
12
, ,…, / 0.674
ˆ5
n
median X X X X X X
= − − −
is sometimes used to estimate the population standard deviation. This estimator is more robust to
outliers than the usual sample standard deviation and usually does not differ much from S when
there are no unusual observations.
(a) Calculate
ˆ
and S for the data 10, 12, 9, 14, 18, 15, and 16.
(b) Replace the first observation in the sample (10) with 50 and recalculate both S and
ˆ
.
SOLUTION
(a)