Find $P(X^2+Y^2<1)$ if $X,Y$ are independent standard normal variables Suppose that $X$ and $Y$ are independent $n(0,1)$ random variables.
(a) Find $P(X^2+Y^2<1)$
My solution:
since then are independent, the $F_{X,Y}(x,y)=F_X(x)F_Y(y)=\dfrac{1}{2\pi} e^{\frac{-(x^2+y^2)}{2}}$
then $P[X^2+Y^2]=\int\int F_{X,Y}$
I am not sure how to define the boundary of the integrals.

If X~N(0,1), then the pdf of $X^2$ is not the square of a normal pdf. Even if you integrate what you have, you'll get the wrong answer. In fact, $X^2$ has a chi-squared distribution with one degree of freedom.
