Question

Suppose that Xi ~ N(, ?2), i-l, are independent. State the distribution of each of the following variables if it is a "named" distribution or otherwise state "unknown." 15. , n and Z1 ~ NO, l), i 1, , k, and all variables

Answer 1

X_i ' N (mu, pi^2), i = 1, ...., n Therefore X_i - mu/sigma ' N (0, 1), i = 1, ....., n (X_i - mu)^2/sigma^2 ' X_1^2, i = 1, ...., n Therefore Sigma^n_i = 1 (X_i - mu)^2/sigma^2 ' X_n^2 Z_i ' N (0, 1), i = 1, ....., k Therefore Z^bar ' N (0, 1/n) let Z_i and Z^bar are not independent Therefore Sigma^k_i = 1 (Z_i - Z^bar)^2 distribution in unknown (m) The distribution of Sigma^n_i = 1 (X_i - mu)^2/sigma^2 + Sigma^n_i = 1 (Z_i - Z^bar)^2 is unknown X^bar ' N (mu, sigma^2/n) Therefore X^bar/sigma^2 ' N (mu/sigma^2, 1/n sigma^2) 0 Sigma^k_i = 1 Z_i ' N (0, 1/K) (n) X^bar/sigma^2 + Sigma^k_1 = 1 Z_i/k ' N (mu/sigma^2, 1/n sigma^2 + 1/k)