Given is the joint density function:
$f_{X,Y}(x,y) = \begin{cases} C & (x-1)^2 + (y-1)^2 \leq 1 \\ 0 & \text{otherwise}\end{cases}$
Obviously C can easily be calculated: $1/\pi$ It's also obvious that this describes a circle with diameter 2 in the x-y-plane (with constant height of $1/\pi$) with it's center at (1,1)
Now the question is to find the conditional probability $P(y \leq x | x \geq 1)$
I think this is solvable by using the bayesian theorem, hence:
$P(y \leq x | x \geq 1) = \frac{P(y \leq x \wedge x \geq 1)}{P(x \geq 1)}$
It's also easy to see that $P(x \geq 1)$ = 1/2
but i can't find out how to calculate $P(y \leq x \wedge x \geq 1)$
Does anyone can give me a hint?