(Q1) Let X1, Xn be random sample drawn from a random variable X with pdf f(x; 0) = S0x0-1, 0

Suppose that X ~ Gamma(3, b), with pdf -br f(r\b) = 2 on a > 0, where b> 0. It is required to test the null hypothesis Họ : b= bo against the alternative hypothesis H1 : b = b1, based on a random sample #1,x2,..., Tn. (a) Find the log-likelihood l(b). (b) Let LR be the likelihood ratio L(b1)/L(bo). Show that the log of the likelihood ratio is log(LR) = 3n log bo (2) - (b1 – bo) ri- i=1 (c) Using the expression for log(LR) that you obtained in part (b), show that the rejection region R= {x : log(LR) > d} for the test is of the form T bo. Hence give c' in terms of d.

34. Consider the continuous random variable X whose pdf is given by f(r) = (a-r³)Io<<1(1). (a) Find the value of a that makes f(z) a pdf. (b) Find E(X). (c) Find V(X). (d) Find P(X ≤).

Suppose Xn is the sample mean of a random sample of size n from a distribution that has a Gamma(2,1/2) (where a = 2, B = }) pdf f(x) = 4xe-2a, 0 < x < ∞, zero elsewhere. Use the Central Limit Theorem to deduce that the random variables Vn(Xn – 1) converges in distribution to N(0, 1/2).