# Joint Normal vs Chi-Square PDF in Distance Testing

In my previous question (Integrating Joint Normal Distribution in R3), I describe a joint normal PDF with three independent random variables of $\mathbf N_3(0,\mathbf I)$. The joint pdf is $p(x,y,z)=p(x)p(y)p(z)={(1/\sqrt {2\pi})}^3e^{-{(x^2+y^2+z^2})/2}$ and the distribution I'm interested in becomes $$P(r<r_o)={(1/\sqrt {2\pi})}^3\int _{x^2+y^2+z^2<r_o^2}e^{-{(x^2+y^2+z^2})/2} dxdydz$$ This can be integrated through change of variables and use of integration by parts to

(Updated/Corrected)
\begin{align} \ 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 \\ & = erf(\frac{r_o}{\sqrt{2}}) - \sqrt{ \frac{2}{\pi}} r_o e^{- \frac{r_o^{2}}{2}} \end{align}

On the other hand, I would expect $P(r<r_o) \equiv P(r^2<r_o^2)$

This is where colleagues expect a degree-3 $\chi^2$ distribution to be applicable. For this I used the formula from Wikipedia for the cumulative distribution of the $\chi^2$ with k=3 for three degrees of freedom:

$$F(x;k)=\frac{\gamma(\frac{k}{2},\frac{x}{2})}{\Gamma(\frac{x}{2})}$$

Using $x=r^2$, I used this Matlab code to model it:

chi2Dist = gammainc(r.^2/2, 3/2,'lower') / gamma(3/2);


A plot comparing the two CDFs is shown below, and apart from the obvious scaling error in the $\chi^2$ CDF, they look like a match. I'd like some ideas on how to show they are the same (or not). (Also, how did I get the scale wrong on the $\chi^2$ CDF?)

• There must be "-" instead of "+" in the second line of $P(r<r_0)$. Also $1/\sqrt{2}$ should not be there, i.e. $$P(r\le r_0)=erf\left(\frac{r_o}{\sqrt{2}}\right) -\sqrt{ \frac{2}{\pi}} r_o e^{- \frac{r_o^{2}}{2}}$$ – d.k.o. Mar 7 '16 at 5:53
• Thanks, I see that now. I'll make the correction. – Jim Mar 7 '16 at 19:48

The reason why you have a scaling problem is because the Matlab function gammainc() is already regularized. Refer to the documentation for this function. You will see that the integral is already divided by $\Gamma(a)$. The function $\gamma(a,z)$ as described in the Wikipedia article is the non-regularized (lower) incomplete gamma function--it isn't divided out.