The continuous random variables X and Y have joint pdf f(x,y) where

f(x,y)=k(x^2 +2xy) for 0 < x < 1 and 0 < y < 1,

i)Find k.

Do i integrate first with respect to x and then w.r.t y? Then do i put the answer equal to 1 and find k? When i do this i get k=6/5.

ii)Find the marginal pdf of X and of Y.

is the marginal pdf of X the integral from 0 to 1 of f(x,y) wrt y?

then the marginal pdf of Y is calculated similarly to this - integral from 0 to 1 of f(x,y) wrt x?

iii)find prob[Y<0.5|X<0.5].

iv)find prob[Y<0.5|X=0.5].

v)find prob[X<Y].

I have a good idea how to do these three questions but its part (vi) that im not sure about

vi)Find E[Y|X=0.5]?

Any help is appreciated! Thanks.!

f(x,y)=k(x^2 +2xy) for 0 < x < 1 and 0 < y < 1,

i)Find k.

Do i integrate first with respect to x and then w.r.t y? Then do i put the answer equal to 1 and find k? When i do this i get k=6/5.

ii)Find the marginal pdf of X and of Y.

is the marginal pdf of X the integral from 0 to 1 of f(x,y) wrt y?

then the marginal pdf of Y is calculated similarly to this - integral from 0 to 1 of f(x,y) wrt x?

iii)find prob[Y<0.5|X<0.5].

iv)find prob[Y<0.5|X=0.5].

v)find prob[X<Y].

I have a good idea how to do these three questions but its part (vi) that im not sure about

vi)Find E[Y|X=0.5]?

Any help is appreciated! Thanks.!

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