Question

(2) Let X, X, be a random sample from normal distribution N (,o2), stribution N(u, a and let S2 be the sample variance: (a) [8pts] show that ES-g? (b) [8pts] For a random sample of size 2 (i.e. n 2), derive that /02 ~ Z2 where Z has standard normal distribution.

Answer 1

Let x_1, x_2, ... x_n be a random sample from Normal distribution N(mu, sigma^2) i.e x_1, x_2, ... x_n N(mu, sigma^2) and Let s^2_n be the sample variance s^2_n = 1/n - 1 sigma^n_i = 1 (x_i - x)^2 = 1/n - 1 sigma^n_i = 1 (x^2_i + x^2 - 2x_i x) = 1/n - 1 [sigma^n_i = 1 x^2_i - nx] (a) Now, ES^2_n = 1/n - 1 middot E[sigma^n_i = 1 x^2_i - nx^-2] = 1/n - 2 [sigma^n_i = 1 E(x^2_i) - n middot E x^-2] As we know, x ~ N[mu, sigma^2/n) So, E s^2_n = 1/n - 1 [n (mu^2 + sigma^2) - n(mu^2 + sigma^2/n)] = 1/n - 1 [n mu^2 + n sigma^2 - n mu^2 - sigma^2] = 1/n - 1 [(n - 1) sigma^2] Therefore E s^2_n = sigma^2 \r\n (b) For a Random sample size size 'z' n =