MATH1315: Assignment 2

1. (a) Let X1, X2, X3 be iid (identical and independently distributed) ran- dom variables with common pdf f (x) = e−x, x > 0 and equal 0 Find the joint pdf of Y1 = X1, Y2 = X1 + X2 and Y3 = X1 + X2 + X3.

• Let X1, X2, . . . , Xn be random sample from a population with pdf fX (x) = 1/θ if 0 < x < θ and 0 Let X(1), X(2), . . . , X(n)be the order statistics.
  • Find the joint pdf of x(1) and x(n).
  • Show that Y1 = X(1)/X(n) and Y2 = X(n) are independent ran- dom v

(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 < x < ∞, zero Use the Central Limit Theorem to deduce that the random variables √n(X¯n − 1) converges in distribution to N (0, 1).
   • Use the Delta method to find the limiting distribution of the random variables √n(√X¯n − 1).
   • Use the limiting distribution of part (b) to find an approximate probability for P (√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, β) distribution, , its probability density function is given by



 xe−

x/β

f (x; β) =

β2        ,     0 ≤ x < ∞

0,      otherwise.

• What is the Maximum Likelihood Estimator (MLE) βˆ of β?
• Show that the estimator βˆ is unbiased for β.
• Prove that βˆ is efficient.
• For the general gamma(α, β) distribution, find the method of mo- ments estimators of α and β.

(4 + 4 + 4 + 4 = 16 marks)