January – Inverse of a Gaussian variable

There is a lot of literature on the inverse of a Gaussian distributed variable (this should not be fixed with inverse Gaussian distribution – it is a different matter). The inverse distribution is ill-behaved; the mean and variance do not generally exist.

I came up with a simple approximation that works well if the mean is large enough and the variance is small enough (I have yet to work out the details of the exact conditions for this approximation. However, the results can be verified, e.g., by Monte Carlo simulations).

First, approximate the Gaussian distributed variable $X \sim \mathcal{N}(\mu, \sigma^2)$ by a log-normally distributed variable $X \approx Y \sim$   Lognormal$(\mu_{\text{LN}}, \sigma_{\text{LN}}),$ with corresponding mean and variance, i.e.,

$$\mu = \exp\left( \mu_{\text{LN}} + \frac{\sigma_{\text{LN}}^2}{2} \right)$$

and

$$\sigma = (\exp( \sigma_{\text{LN}}^2) - 1)\exp(2 \mu_{\text{LN}} + \sigma_{\text{LN}}^2).$$

We leave the solving of $\mu_{\text{LN}}$ and $\sigma_{\text{LN}}$ as an easy exercise for the reader. (:

Using the theory of log-normal distribution, the inverse of $X$ is now given by

$$1/X \approx 1/Y \sim$$   Lognormal$$(-\mu_{\text{LN}}, \sigma_{\text{LN}}).$$

That's it!

References:

• Log-normal distribution
• Díaz-Francés, Eloísa; Rubio, Francisco J. (2012-01-24). "On the existence of a normal approximation to the distribution of the ratio of two independent normal random variables". Statistical Papers. Springer Science and Business Media LLC.