Suppose I have a Cumulative Distribution Function like this:
$F(x,y)=\frac { (x\cdot y)^{ 2 } }{ 4 } $
where $0
And I want to find the probability of $P(X+Y<1)$.
Since $x<1-y$ and $y<1-x$, I plug these back into the CDF to get this:
$F(1-y,1-x)=\frac { ((1-y)\cdot (1-x))^{ 2 } }{ 4 } $
Because of the constraint where $0
$\int _{ 0 }^{ 1 }{ \int _{ 0 }^{ 2 }{ \frac { ((1-y)\cdot (1-x))^{ 2 } }{ 4 } dxdy } } =\frac { 1 }{ 18 } $
This answer, however, is incorrect. My intuition for doing this is that because the two variables are somewhat dependent on each other to maintain the inequality of less than $1$, I want to "sum"(or integrate) all the probabilities within the possible range of values of $x$ and $y$ that satisfy the inequality. Somehow, the answer, which is $\frac{1}{24}$, doesn't seem to agree with my intuition.
What have I done wrong?