Density of the product of two real gaussian random variables

Matteo Frigo

February 9, 2002

Lemma 1 Let a and b be independent real normal random variables with mean 0 and variance 1. The probability density function of |ab| is f(c) = K_{0}(|c|)/π, where K_{0} is the modified Bessel function of second kind and index 0.

Proof: In the following, assume for simplicity that c ≥ 0. (Because f(c) is an even function, the case c < 0 follows by symmetry.) We have

Pr [c ≤ ab ≤ c + dc] = \frac{1}{\sqrt{2π}} ∫_{−∞}^{∞} e^{−a^{2}/2}Pr [\frac{c}{|a|} ≤ b < \frac{c + dc}{|a|}] da
= \frac{dc}{2π} ∫_{−∞}^{∞} e^{− \frac{1}{2} (a^{2}+ \frac{c^{2}}{a^{2}} )} \frac{da}{|a|}
= \frac{dc}{π} ∫_{0}^{∞} e^{− \frac{1}{2} (a^{2}+ \frac{c^{2}}{a^{2}} )} \frac{2a da}{2a a}
[let a^{2} = u, 2a da = du]
= \frac{dc}{π} ∫_{0}^{∞} e^{− \frac{c}{2}(\frac{u}{c} + \frac{c}{u})} \frac{du}{2u}
[let u = ce^{z}, du/u = dz]
= \frac{dc}{π} ∫_{−∞}^{∞} e^{−c cosh z} \frac{dz}{2}
= \frac{dc}{π} ∫_{0}^{∞} e^{−c cosh z} dz
= \frac{K_{0}(c) dc}{π} .

(For the latter identity, see [1, 9.6.24].)

References

[1] M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover, New York, 1965.