let the continuous random vari
let the continuous random variables X and Y have the jointpdf:
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 isfY|X(y|x),
iii) E(Y|x) and Corr(X,Y).
Answer:
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
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