# Integrating Joint Normal Distribution in R3

With the multivariate distribution of $\mathbf N_3(0,\mathbf I)$, I want to determine the probability $\rho$ within the sphere defined by a vector of length $r_o$.

I start with the joint pdf being $p(x,y,z)=p(x)p(y)p(z)={(1/\sqrt {2\pi})}^3e^{-{(x^2+y^2+z^2})/2}$.

To integrated this, I have $$P(r<r_o)={(1/\sqrt {2\pi})}^3\int _{x^2+y^2+z^2<r_o^2}e^{-{(x^2+y^2+z^2})/2} dxdydz$$ At this point, I thought I would do a change in variables and integrate the volume of the spherical shells:

$$P(r<r_o)={(1/\sqrt {2\pi})}^3 \int _{0 \le r \le r_o} 4\pi r^2 e^{-r^2/2} dr$$

I'm not sure how to proceed from here (or if I made a wrong turn along the way).

Also, this is related to performing a distance hypothesis test, and I want to know if anything related to the chi-square distribution falls out of this integral.

So far so good. Next step could be integration by parts ($u = r$, $dv = r e^{-r^2/2}$).