Q4. For any distribution it is usual convention to denote u, and u, are the r" moment of the population about the arbitrary value and about the mean, then m, and m, denotes the corresponding moments of sample. If n is the size of the sample, then show that i) E (m, ) = Hr ii) Var (m, ) = = [Azr - (u, )2] iii) Cov( m,, m's) = = [Arts - Mrus] iv ) E (m2 ) = ( 1 - 1 ) / 2 v) Var (m2) = 2 (1 -1)" #4 - (-1)(n-3) 72 2 = [M4 - 143] to order 1 and that for normal parent Var(m2) = 204 and Var(s2) = 20*/(n -1) vi) Var(m,) =? vii) Cov(m,, m,) =? Q5. If yi's, i = 1, 2, 3, .., k are k function of random variables calculated from a sample of size n and if each y; is distributed about mean 1; with variance of order 1, then prove that Var (f) = > var (v.) + [ (ox,) (ox, ) Cov(v.>;) i=1 itj =1 where, f = f()1, 2. ..., VK) is any function of y's and possesses partial derivatives