$D=\{ (x,y):0\leq y\leq x\leq 1\}$find $f_{X,Y}(x,y)$ and $P(X\leq a, Y\leq b)$

Question

$D=\{ (x,y):0\leq y\leq x\leq 1\}$

I have to find $f_{X,Y}(x,y)$ and $P(X\leq a, Y\leq b)$

I found $f_{X,Y}(x,y)=2$ (this is correct) , but I can't find $P(X\leq a, Y\leq b)$.

My solution:

$ \displaystyle P(X\leq a, Y\leq b) = \int_0^a dx \int_{0}^b2dy=2ab~$ but the correct answer is $b(2a-b)$ , where am I wrong?

Thanks !

Answer 1

You are missing that $D=\{ (x,y):0\leq y\leq x\leq 1\}$ so considering $b \lt a$,

$P(X \lt a, Y \lt b)$ $ ~~= \displaystyle \int_0^b \left[ \int_0^x f_{XY}(x,y) ~ dy\right] ~ dx + \int_b^a \left[\int_0^b f_{XY}(x,y) ~ dy \right]~ dx$

If $f_{XY}(x, y) = 2$, we do get $P(X \lt a, Y \lt b) = b(2a-b)$

Here is a diagram that may help understand -