Question

Random variables X and Y are distributed according to the joint PDF

fX,Y(x,y) = {ax,0,if 1≤x≤2 and 0≤y≤x,otherwise.}

1. Find the constant a.

2. Determine the marginal PDF fY(y). (Your answer can be either numerical or algebraic functions of y).

For 0≤y≤1, fY(y)=

For 1<y≤2, fY(y)=

3. Determine the conditional expectation of 1/X given that Y=3/2.

E[1/X∣Y=3/2]=

Answer 1

1. a= 0.4286

2.
For 0≤y≤1, fY(y)= 0.6429

For 1<y≤2, fY(y)= 0.2142*(4-y^2)

3. E[1/X∣Y=3/2]= 0.5714