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.

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.

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