SOA 115 questions!!!

[eric1990710]

It's a coincident, the fact that x to x+1 us equal to 0 to 1 when the x got canceled out. Also i think there is a much better approach for this question. Since Y is a constant therefore, y must be a uniform. And we are given x<y<x+1. Continous uniform variance= (b-a)^2/12, so (x-1-x)^2/12=1/12

[ronaldy27]

Hi I've looked up other threads for SOA 115 but no one brought up this question so I am going to ask here.
The joint density function of x and y is 2x for 0<x<1 and x<y<x+1.
The questions wants us to find the conditional variance of Y given X=x.

You can quickly find that the conditional density of Y given X=x is 1.
Now we know from one of the assumptions that x<y<x+1.
However, I realized that 0<y<1 because the conditional density is 1 and this is true if the conditional density of y given X=x is uniform defined in (0,1).
Finally, my question is this: the solution finds the conditional variance by integrating from x to x+1 to to find the first and second moment of y given X=x. The answer is 12. I get the same answer if I integrate instead of x to x+1 but from 0 to 1. This works or is it a coincidence that I got the same answer.