If X,Y,Z are independent standard normal random variables, find the distribution of X^2 + Y^2 + Z^2

Question

If $X,Y,Z$ are independent standard normal random variables, find the distribution of $X^2 + Y^2 + Z^2$. Using spherical coordinates.

My doubt appears in that last sentence, how can I use spherical coordinates?

Answer 1

The distribution of $X^2 + Y^2 + Z^2$ is uniquely determined by the cumulative distribution function it corresponds. To find this, we must find $$P(X^2 + Y^2 + Z^2 \leq t)$$ for some constant $t$. From here, it depends on how you parametrize spherical coordinates in your course. I am accustomed to taking $\phi$ from $- \frac{\pi}{2}$ to $\frac{\pi}{2}$, though some people take it from $- \pi$ to $\pi$. In the way that I'll be doing it, we must account for a factor of $r^2 \cos \phi$.

Since $x^2 + y^2 + z^2 = r^2$ in spherical coordinates, $r$ will range from $0$ to $\sqrt{t}$. Then $\theta$ ranges from $0$ to $2\pi$ and $\phi$ from $- \frac{\pi}{2}$ to $\frac{\pi}{2}$. The integral is then: $$P(X^2 + Y^2 + Z^2 \leq t) = \int_{\theta = 0}^{2\pi} \int_{\phi = - \frac{\pi}{2}}^{\frac{\pi}{2}} \int_{r=0}^{\sqrt{t}} f_{X,Y,Z} (\mathbf{\vec{v}}) r^2 \cos \phi \; dr d\phi d\theta,$$ where $f_{X,Y,Z} (\mathbf{\vec{v}})$ is the joint density of $X$, $Y$, and $Z$. Since $X$, $Y$, and $Z$ are independent, this is the product of the marginal densities. This may lookg rather messy at first, but the conversion to spherical cleans up the integrand quite nicely.

This will give the CDF of $X^2 + Y^2 + Z^2$, from which you can read off its distribution. I'll let you finish it from here.

What you'll find eventually is that this distribution is chi-square with three degrees of freedom. Generally speaking, $$X_1^2 + X_2^2 + \ldots + X_n^2 \sim \chi(n)$$ where the $X_i$ are i.i.d. standard normal random variables.