Question

The joint probability density function of random variables X and Y is given by f(x,y) ={10xy^2 0≤x≤y≤1,0 otherwise.

(a) Compute the conditional probability f_X|Y(x|y).

(b) Compute E(Y) and P(Y >1/2).

(c) Let W=X/Y. Compute the density function of W.

(d) Are X and Y independent? Justify briefly.

Answer 1

(a)   

(b) the marginal probability density function of Y: Also, substituing the value of c,

(c) the conditional probabiltiy density function of X given Y=1:

(d)Pr {X=Y}: Setting limits of Y from 0 to X i.e., 0 < t < s, we get

(e) Pr{X+Y<1} = Pr (Y < 1-X )

(f) check whether X and Y are independent:

If conditional is equivalent to marginal then X and Y are independent.

i.e.

We already found f(Y), Now we need to find f(X):

Hence X and Y are independent.