# The random variables X Y f XY xyax^xyx 0 otherwise Find the constant a

and are distributed according to the joint PDF

,(,) = {2 if 1≤≤2 and 0≤≤,

- Determine the marginal PDF
*fY*(*y*)

a. If 0≤*y*≤1

b. if 1<*y*≤2

- Determine the conditional expectation of 1/((
*X^*2)*Y), given that*Y*=5/4

I do not need the answer but hints. I already answered question 1.

Question 2:

You have to integrate $f_{X,Y}(x,y)$ with respect to x to get the marginal PDF of Y. The range of x is tricky. But we know that $0≤y≤x$ . So $y≤x$ and also $0≤x≤2$ . Combining the two gives $y≤x≤2.$ So integrate $f(x,y)$ with respect to x with the limits y to 2.

Also remember that since in both a and b $y≤2$ , the answer will be the same.

Question 3:

You can substitute $Y=45 $ in the expression $X_{2}Y1 $ . Then,

$E[X_{2}Y1 ∣Y=45 ]=∫_{45}5x_{2}4 ⋅f_{Y}(Y=45 )f_{X,Y}(x,45 ) $ $dx$. The denominator we have from part 2. This should be a simple integration now.