Question

1. Show that X and Y are independent if and only if the conditional distribution of X given Y = y is the same as the marginal distribution of X for all y. When X and Y are independent show that E(X Y = y) = EX.

Answer 1

Solution - Show that iff Xendy are independent ) f(y) = f(x). so If Xand y are independent Rus then then the since f(ly= f(19) fly) Xona y are Indspondent =f(1).f44 I then | 44. y) = f(21f44) 1 Flily) = f(x) and also given that when X and y are I indepent then show that E (X[v=JJ = E(X-

I taking Lots Expx=y)= fè.£49 de cw=få frente (2/4) de Elx)=fx ftridae = f(4. f4 de - f(214)=f(1,92 ---Fly) fand it . X and Yare Independent =s riflegle then f(x,y)=fly fly) E[X[=yF E () Henco poved,