Section A
Q1.
(a) A printing machine can print n "letters", say α_1, α_2, ..., α_n. It is operated by electrical impulses, each letter being produced by a different impulse. Assume that p is the constant probability of printing the correct letter and the impulses are independent. One of the n impulses, chosen at random, was fed into the machine twice and both times the letter α_1 was printed. Compute the probability that the impulse chosen was meant to print α_1.
(b) A positive integer X is selected at random from the first 50 natural numbers. Calculate PX + (48)/(X) > 26).
(c) Using Central Limit Theorem, show that limn→∞∑k=-∞5n\binom5nk1/5)^k4/5)^5n-k = 1/2.
(d) Let X1 and X2 be iid Poisson (λ) variates. Examine whether the (i) statistic T = X1 + 2X2 is sufficient for λ. Let X be Poisson (λ) and \psi(λ) = e-3λ. Show that (-2)X is an (ii) unbiased estimator for the parametric function \psi(λ) and examine whether it is a reasonable estimator.
(e) Establish the necessary and sufficient condition for an unbiased estimator to be UMVUE.
