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 ($u = r$, $dv = r e^{-r^2/2}$).

  • $\begingroup$ Thanks, that works out great i numerical analysis once I got the scaling right for the term containing an erf function. $\endgroup$ – Jim Mar 5 '16 at 20:08

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.