Section A
Q1.
(a)
An unbiased six-sided die is thrown twice. Let X denote larger of the scores obtained. Then show that the probability mass function (p.m.f.) is given by
pX(x) = (2x-1)/(36), x = 1, 2, ..., 6 = 0, elsewhere
(b)
If X follows the binomial (n, p), then for any a > 0, show that
P\[(X)/(n)-p≥ a\]≤ (1)/(4n a2)
(c) Let Xk, k = 1, 2, 3, \dots be i.i.d. exponential random variables with mean λ. Find the limiting distribution of first-order statistic X(1) of sample of size n.
(d)
Let X belong to an exponential family of distributions of the form
fX(x, θ) = ex log θ + log g(θ) + w(x)
Obtain maximum likelihood estimation equation based on n observations. Solve the same for Poisson distribution with variance θ.
(e)
Find a uniformly minimum variance unbiased estimator for the parameter λ based on a random sample of size n drawn from the distribution
f(x, θ) = \begincases √((λ)/(π)) x-1/2 e-xλ, & x > 0; λ > 0 \\ 0 & , \text otherwise \endcases
