Am I obtaining the Incorrect Equation for this Joint Probability density Function?

Question

I am confused over how I am messing the equation up to solve this problem listed below

I was asked in Part C to find P(X > Y).

Therefore I made a double integral as such:

The answer to the problem is 15/16, but why I am I getting -1 by doing it this way?

Am I calculating the equation incorrectly?

Answer 1

Your integration domain is just wrong, you don't consider the restriction $X < 1$, e.g. $y = \frac{3}{2}$ is part of your domain and for this the inner integral becomes negative. Nevertheless $y = \frac{3}{2}$ is not valid for the domain $\{X > Y\}$

So your double integral should look like $$\int_0^1 \int_y^1 \ldots$$ because $X>Y$ implies $y < 1$.