Math1315 Let X1 X2 X3 Be Iid Identical And Independently Distributed Random Variable 4140567

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

n – 1).
(c) Use the limiting distribution of part (b) to find an approximate
probability for P(
v

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) ߈ of ß?
(b) Show that the estimator ߈ is unbiased for ß.
(c) Prove that ߈ is efficient.
(d) For the general gamma(a, ß) distribution, find the method of moments estimators of a and ß.

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