Inverse of sum of non-zero mean squared random variables

Question

I am looking for a distribution $D$ with the following property.

Suppose $X_i \sim D$, $E[X] \geq 1$, and $Y \sim \frac{1}{\sum_i X_i^2}$. I would like the distribution $D$ and the distribution $Y$ to have closed-form (and well-understood) PDFs.

I know that if $D$ is a standard normal, then $Y$ will be distributed like an inverse gamma. However, I'd like the expectation of $X$ to be positive. Anyone know of a pair of distributions like this?

Answer 1

• Let $X \sim \text{ChiDistribution}(v)$ where $E[X] = \frac{\sqrt{2} \Gamma \left(\frac{v+1}{2}\right)}{\Gamma \left(\frac{v}{2}\right)}$ and pdf $\phi(x)$:

$$\phi(x) = \frac{2^{1-\frac{v}{2}}}{\Gamma \left(\frac{v}{2}\right)} e^{-\frac{x^2}{2}} x^{v-1} \quad \text{for } x > 0$$

• Then $W = X^2 \sim \text{Chisquared}(v)$.
• Sum of $n$ iid Chisquared rvs: $\quad S = \sum _{i=1}^n W_i \sim \text{Chisquared}(n v)$
• If $S$ is $\text{Chisquared}(n v)$, then $Y = \frac{1}{S} \sim \text{InverseChisquared}(n v)$ with pdf $f(y)$:

$$f(y) = \frac{ y^{-\frac{n v}{2}-1} \exp \left(-\frac{1}{2 y}\right)}{2^{\frac{n v}{2}} \Gamma \left(\frac{n v}{2}\right)} \quad \quad \text{for } y > 0$$

This satisfies all your requirements (in particular, if $v>\sqrt{2}$, then $E[X] >1$).