Suppose that the joint chance density of two random variables X and Y is given by:
f(x,y) = half, 0 < y < x < 2
First, I attempted discovering marginal densities f(x) and f(y)
$ f_1 (x)= ∫_0^x〖half dy〗 = [1/2y]^x_0 = 1/2x,$ for 0 < x < 2
$ f_2 (y)= ∫_y^2〖half dx〗 = [1/2x]^2_y = 1 – 1/2y, $ for 0 < y < x
Then I went on to seek out E(X) and E(Y):
$ E(X) = ∫^∞_∞〖xf(x,y) dx〗=∫_0^2〖(half x^2 )dx= 〗 1/6 x^3]^2_0 = 4/3 $
$ E(Y) = ∫^∞_∞〖yf(x,y) dy〗=∫_0^x〖(y-1/2 y^3 )dy $
$ = 〗 half y^2-1/eight y^4]^2_0 = half x^2-1/eight x^4 = 3/2 $
Then, I calculated E(XY):
$ E(XY) = ∫^∞_∞∫^∞_∞〖xy f(x,y) dx dy〗 = ∫^2_0∫_y^2〖half xy〗 dx dy$
= $∫_0^2〖[1/4 x^2y 〗 ]_y^2 dy = ∫_0^2 y-1/4y^Three dy = [1/2y^2-1/16y^4]^2_0 = 2-1 = 1 $
Lastly, Cov(X,Y) = E(XY) – E(X)E(Y) = 1 – (4/3 * 3/2) = 1 – 2 = -1
Is that this right?
I actually have no idea if I’m proper and there’s no supply for me to verify my reply. Please assist.