MATH1315: Assignment 2

1. (a) Let X1, X2, X3 be iid (identical and independently distributed) random variables with common pdf f(x) = e

-x

, x > 0 and equal 0

elsewhere. Find the joint pdf of Y1 = X1, Y2 = X1 + X2 and

Y3 = X1 + X2 + X3.

(b) Let X1, X2, . . . , Xn be random sample from a population with pdf

fX(x) = 1/? if 0

be the order statistics.

(i) Find the joint pdf of x(1) and x(n)

.

(ii) Show that Y1 = X(1)/X(n) and Y2 = X(n) are independent random variables.

(Hint:First derive the joint pdf of Y1 and Y2, say h(y1, y2), by using the

defined transformation functions and the the joint pdf of X(1) and X(n)

which obtained from (i), then show that the joint pdf h(y1, y2) can be

factorized as a product of a function of y1 and a function of y2.)

(5 + 7 = 12 marks)

2. (a) Suppose X¯

n is the sample mean of a random sample of size n from

a distribution that has an exponential pdf f(x) = e

-x

, 0

zero elsewhere. Use the Central Limit Theorem to deduce that the

random variables v

n(X¯

n – 1) converges in distribution to N(0, 1).

(b) Use the Delta method to find the limiting distribution of the random

variables v

n(

v

X¯

n – 1).

(c) Use the limiting distribution of part (b) to find an approximate

probability for P(

v

X¯

36 = 1.25).

(6 + 5 + 5 = 16 marks)

3. Let X1, X2, . . . , Xn represent a random sample from a population with a

gamma(2, ß) distribution, ie., its probability density function is given by

f(x; ß) =

?

??

??

xe-x/ß

ß2

, 0 = x

0, otherwise.

(a) What is the Maximum Likelihood Estimator (MLE) ßˆ of ß?

(b) Show that the estimator ßˆ is unbiased for ß.

(c) Prove that ßˆ is efficient.

(d) For the general gamma(a, ß) distribution, find the method of moments estimators of a and ß.

1. (a) Let X1, X2, X3 be iid (identical and independently distributed) random variables with common pdf f(x) = e

-x

, x > 0 and equal 0

elsewhere. Find the joint pdf of Y1 = X1, Y2 = X1 + X2 and

Y3 = X1 + X2 + X3.

(b) Let X1, X2, . . . , Xn be random sample from a population with pdf

fX(x) = 1/? if 0

be the order statistics.

(i) Find the joint pdf of x(1) and x(n)

.

(ii) Show that Y1 = X(1)/X(n) and Y2 = X(n) are independent random variables.

(Hint:First derive the joint pdf of Y1 and Y2, say h(y1, y2), by using the

defined transformation functions and the the joint pdf of X(1) and X(n)

which obtained from (i), then show that the joint pdf h(y1, y2) can be

factorized as a product of a function of y1 and a function of y2.)

(5 + 7 = 12 marks)

2. (a) Suppose X¯

n is the sample mean of a random sample of size n from

a distribution that has an exponential pdf f(x) = e

-x

, 0

zero elsewhere. Use the Central Limit Theorem to deduce that the

random variables v

n(X¯

n – 1) converges in distribution to N(0, 1).

(b) Use the Delta method to find the limiting distribution of the random

variables v

n(

v

X¯

n – 1).

(c) Use the limiting distribution of part (b) to find an approximate

probability for P(

v

X¯

36 = 1.25).

(6 + 5 + 5 = 16 marks)

3. Let X1, X2, . . . , Xn represent a random sample from a population with a

gamma(2, ß) distribution, ie., its probability density function is given by

f(x; ß) =

?

??

??

xe-x/ß

ß2

, 0 = x

0, otherwise.

(a) What is the Maximum Likelihood Estimator (MLE) ßˆ of ß?

(b) Show that the estimator ßˆ is unbiased for ß.

(c) Prove that ßˆ is efficient.

(d) For the general gamma(a, ß) distribution, find the method of moments estimators of a and ß.

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