conditional probability with joint pdf

Question

So I got stuck at the following problem: Theres a bivariate random variable with joint pdf $$f_{X,Y}(x,y)=\frac{e^\frac{-(x^2+y^2)}{2\sigma}}{2\pi\sigma^2}$$ $$(-\infty\lt x,y\lt \infty)$$ I was asked to find $P(X,Y)$ while given$\quad$ $x^2+y^2\le a^2$. please help me ! How do I set up the boundaries for my integral when calculating the CDF???

Answer 1

It is not clear what you mean. Given pdf $f_{X,Y}$, pdf conditional on $x^2+y^2\leq a^2$ is $f_{X,Y|X^2+Y^2\leq a^2}=\frac{f_{X,Y}}{\mathbb{P}[X^2+Y^2\leq a^2]}$.

You mean how do you calculate $\mathbb{P}[X^2+Y^2\leq a^2]$? Just observe $x^2+y^2\leq a^2$ is a circle with center at $(0,0)$ and radius $a^2$. So you take $x\in[-a,a]$ and for each $x$ you take $y\in[-\sqrt{(a^2-x^2)},\sqrt{(a^2-x^2)}]$.