Multivariate Data Analysis — DU B.A. (Prog.) with Statistics as Major Discipline Previous Year Question Papers

DSE: Multivariate Data Analysis B.A. / B.Sc. (P) under NEP-UGCF Attemp.t six questions in all, selecting Three from each section. 1. (a) For the bivariate normal distribution`: f(x,y)=fegxp(-±[(x+2)2-2.8(x+2)07-1)+4fy'-1)2]); obtain (i) the parameters of the distribution (ii) the mean and variance of the condifionaldistributionofY|X=5. (b) For (X , y)~ BVIVQti, #2 , crt2, cr22, p), prove that X and y are independent iff p = 0. 2.(a)Derivethemomen{generating.function for(X,y)~BVIV(#[,#2,a:12,o22,..a). (b) I.f xi and X2 are Standard normal variates with correlation coefficient p between them then show that correlation coefficient between xf and x2 is p2. (8;7) 3. (a) If x and Y are standard normal variates with coefficient of correlation p between them, then obtain the distribution of jfz+yz-2pxy a) If (X , ,y)~ BVIV(4ti, #2 , a.12, al22, p), then find the correlation coefficient between ex and ey. (8,7) 4. (a) If (X,y) ~ BW(#, , #2, a.,2, cr3 ,p) then showthat linear combination of ,X and yis a normal variate. Prove that converse also holds good. q]) Ifx have a p-variate normal distribution IVp (#,I:) and D is p x p non-singular matrix then sho" that D x ~ Np{Dpr, DED'}. (8,]) 5. (a) If A have a p-vczrz.ore normal distribution JVpQ;',I) and if X = (Xi,X2), where Xi is a (q x 1) and X2is a (® - q) X 1)` 'sub-vector, obtain conditional distribution of Xi giLven JY2 = xz.,Hence show that (i) the mean of Xi given X2 = #2 is linear function of given X2 (ii) covariance matrix.of x] given X2 = x2 does not dependent on x2. (b) Let Xi,X2 and X3 are three variables measured from their respective means, obtain the equation of regression plane.X3 on xi and x2. (8,7) 6. (a) Derive the moment generating function of X , where X ~ IVp tr, I) . (b) Let X follows IV3,(0, Z) `with I = I.; .:; .:I value of p such that (Xi + jf2 + X3) and (Xi ~ X2 - X3) are independent. Princii>al Co`mponents. What are its uses. @) Obtain principal component of z-I: -:2 -:2]. Also, calculate the proportion of the total population variance explained by the first principal component. 8. Describe an orthogonal factor model stating clearly the underlying assumption. Prove that the covariance structure of such a model is expressed by I = LL' + ap, defining clearly the ith communality and the specific variance. (15)