Question

let the continuous random variables X and Y have the joint pdf:

f(x,y)=6x , 0<x<y<1

i) find the marginal pdf of X and Y respectively,

ii) the conditional pdf of Y given x, that is f_Y|X(y|x),

iii) E(Y|x) and Corr(X,Y).

Answer 1

i)

The range of x is (0, y)

The range of y is (x, 1)

The marginal PDF of X is,

for 0 < x < 1

The marginal PDF of Y is,

for 0 < y < 1

ii)

for x < y < 1

iii)

For Corr(X,Y),

Cov(X, Y) = E(XY) - E(X) E(Y) = (2/5) - (1/2) * (3/4) = 1/40

Var(X) = E(X²) - [E(X)]² = (3/10) - (1/2)² = 1/20

Var(Y) = E(Y²) - [E(Y)]² = (3/5) - (3/4)² = 3/80

Corr(X, Y) = Cov(X, Y) /

= (1/40) /

= 0.5773503