January – Inverse of a Gaussian variable

There is quite a pile of literature on the subject of the inverse of a Gaussian distributed variable (this should not be fixed with inverse Gaussian distribution – it is a different matter). In fact, the inverse distribution is ill-behaved; the mean and variance does 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 not worked out the details of the exact conditions for this approximation. However, the results can be verified 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.

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


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

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

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

That's it!