# Logit-normal distribution

Notation Probability density function Cumulative distribution function $P( \mathcal{N}(\mu,\,\sigma^2) )$ σ2 > 0 — squared scale (real), μ ∈ R — location x ∈ (0, 1) $\frac{1}{\sigma \sqrt{2 \pi}}\, e^{-\frac{(\operatorname{logit}(x) - \mu)^2}{2\sigma^2}}\frac{1}{x (1-x)}$ $\frac12\Big[1 + \operatorname{erf}\Big( \frac{\operatorname{logit}(x)-\mu}{\sqrt{2\sigma^2}}\Big)\Big]$ no analytical solution $P(\mu)\,$ no analytical solution no analytical solution no analytical solution

In probability theory, a logit-normal distribution is a probability distribution of a random variable whose logit has a normal distribution. If Y is a random variable with a normal distribution, and P is the logistic function, then X = P(Y) has a logit-normal distribution; likewise, if X is logit-normally distributed, then Y = logit(X)= log (X/(1-X)) is normally distributed. It is also known as the logistic normal distribution,[1] which often refers to a multinomial logit version (e.g. [2] [3] [4] [5]).

A variable might be modeled as logit-normal if it is a proportion, which is bounded by zero and one, and where values of zero and one never occur.

## Characterization

### Probability density function

The probability density function of a logit-normal distribution is:

$f_X(x;\mu,\sigma) = \frac{1}{\sigma \sqrt{2 \pi}}\, e^{-\frac{(\operatorname{logit}(x) - \mu)^2}{2\sigma^2}} \frac{1}{x (1-x)}, \quad x \in (0, 1)$

where μ and σ are the mean and standard deviation of the variable’s logit (by definition, the variable’s logit is normally distributed).

The density obtained by changing the sign of μ is symmetrical, in that it is equal to f(1-x;-μ,σ), shifting the mode to the other side of 0.5 (the midpoint of the (0,1) interval).

### Moments

The moments of the logit-normal distribution have no analytic solution. However, they can be estimated by numerical integration.

### Mode

When the derivative of the density equals 0 then the location of the mode x satisfies the following equation:

$\operatorname{logit}(x) = \sigma^2(2x-1)+\mu .$