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≤,

  1. Determine the marginal PDF fY(y)

a. If 0≤y≤1

b. if 1<y≤2

  1. 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.

Answer & Explanation
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Question 2:

You have to integrate fX,Y(x,y) with respect to x to get the marginal PDF of Y. The range of x is tricky. But we know that 0yx . So yx and also 0x2 . Combining the two gives yx2. So integrate f(x,y) with respect to x with the limits y to 2.

Also remember that since in both a and b y2 , the answer will be the same.

Question 3:

You can substitute Y=45 in the expression X2Y1 . Then,

E[X2Y1Y=45]=4525x24fY(Y=45)fX,Y(x,45) dx. The denominator we have from part 2. This should be a simple integration now.