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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

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

−

−

=

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

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)

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