P 1. (a) Let XXX, be p jointly distributed random variables. Derive the multiple correlation coefficient of X on X. X. X, . What will be the expression of this correlation coefficient when the true regression of X on X. X. linear? (a) Let X. X. X, (k2) jointly follow a multinomial distribution with parameters (n. P1 P2 P ). Derive the marginal distribution of (X1, X2). 2. (a)Let X1, X2, (n. P- P2- X, is P X, jointly follow a multinomial distribution with parameters P). Derive the conditional distribution of X given X = X. X = X3, = Xx. Hence show that the regression equation of Xon X2, X3, X, is linear. (b) Let X = (X1, X2, ..., Xp)' be distributed according to Np (u, ) with joint PDF f(x)=ke-A(x), Derive the expressions for k and A. 3. (a) Let X. X. X and I. Y be independent random samples from continuous distribution functions F and G respectively. Describe a procedure for testing Ho: F(2) = G(2)Vz in terms of the location parameter against all possible alternatives. (b) What is meant by "Discriminant Analysis"? [(8 4. Suppose you have the data on scores (in a 100-point scale) given by 20 students of school A, 20 students of school B and 15 students of school C regarding the COVID protocol followed in the schools. How will you test whether the median score of the three schools differ significantly? 5. Let X. X. X, be p jointly distributed random variables. Explain how you may explain a percentage 'K' of the total variability of X. X..... X, through a number of linear combinations of X, X..... X,. 6. Derive the Moment Generating Function of a multivariate normal distribution.