This question will walk you through computing the correlatio

This question will walk you through computing the correlation of two random variables with a joint density function, Let X and Y have joint density function f(x,y) = For this question recall that if X and Y have joint density f (x,y) then for any function g (x, y) the expectation E [g (X, y)] = g (x, y) f (x, y) dxdy. Show that E[X] = 0. Don\'t argue by symmetry, compute the actual integral. Argue that E[Y] = 0. Show that S.D. (X) = 1/3 Argue that S.D.(y) = Show that Cov(A, Y) = 1/4 Show that (x,y) = 0.75.

Solution

E(x) = 1/2(x) -1/2(x) =0

b) E(X^2) = 1/3(1/2)+1/3(1/2) = 2/6 = 1/3

HEnce vari(x) = 1/3-0 = 1/3

Std dev(x) = 1/rt 3

c) Cov (x,y) = E(XY)-E(x)E(Y) = E(xY) = 1/2(1/2) = 1/4

d) Corr (x,y) = 1/4/1/3 = 3/4 =0.75