Question

I just need to know the answer of question 20. please do do not answer 16 and 19. I have got the answer.

16) X & Y have joint pdf f(x, y) = 3xy, 0 < x <1, x < y < 2 − x Determine the marginal pdf of X on the interval (0, 1)

the answer is 6x – 6x^2

19) Suppose X & Y have a uniform (flat) pdf on the support of problem (16). Determine P(Y > 2X).

the answer is 2/3

20) Continuing problem (19), determine Var(Y | X = .75) a) 1/4 b) 1/12 c) 1/16 d) 1/24 e) 1/48

Answer 1

20)

here f(y|X=0.75)=f(x,y)/f(x) =3xy/(6x-x²)

hence f(y|x=0.75)=3*0.75*y/(6*0.75-0.75²) for 0.75 <y<2-0.75

or f(y|x=0.75)=2y   0.75 <y<1.25

hence E(Y|X=0.75)=integrate(y*f(y),y=0.75..1.25)=1.0208

E(Y²|X=0.75)=1.0625

hence Var(Y|X=0.75)=E(Y²|X=0.75)-(E(Y|X=0.75))² =0.0204

option E is correct