Find $P(X^2+Y^2<1)$ if $X,Y$ are independent standard normal variables

Question

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.

Answer 1

The integral is over the open unit disk, hence if you switch to polar coordinates the result is $$ \frac{1}{2\pi}\int_0^{2\pi}\int_0^1e^{-\frac{r^2}{2}}\;rdrd\theta$$

Answer 2

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.