There is a circle with radius is 1. Then one wants to throw a dart to the circle. Let's assume that dart hit the circle always. But the probability to hit is same to all area. (Uniform Distribution) We want to think this circle with coordinates. That is, $x^2+y^2 = 1$.
In this case, I want to get the density function $f(x,y)$ and also $F(x,y)$ when cumulative density function is $F(x,y)=P(X\leq x $ and $Y\leq y)$.
For example, $F(X\leq 0$ and $Y\leq 0)=1/4$, and $f(X=0$ and $Y=0)=0$.
Could you get the density function of this problem? I have been thought of this for a long time.
Thank you.
1) updated by some comments. Thanks! But how I can derive cdf here?
2) I am not convinced that why $1/\pi$ is pdf here. From this one, it is hard to get the cdf for me. The reason is that probability function is kind of shape of rectangle but $1/\pi$ is related to marginal increase of shape of circle! That is, it is the change rate of radius not change rate of x and y (rectangle)! Could you prove this one?? WHAT is Cumulative Density Function here???