Logit-normal distribution

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Probability density function
Plot of the Logitnormal PDF
Cumulative distribution function
Plot of the Logitnormal PDF
Notation P( \mathcal{N}(\mu,\,\sigma^2) )
Parameters σ2 > 0 — squared scale (real),
μR — location
Support x ∈ (0, 1)
pdf \frac{1}{\sigma \sqrt{2 \pi}}\, e^{-\frac{(\operatorname{logit}(x) - \mu)^2}{2\sigma^2}}\frac{1}{x (1-x)}
CDF \frac12\Big[1 + \operatorname{erf}\Big( \frac{\operatorname{logit}(x)-\mu}{\sqrt{2\sigma^2}}\Big)\Big]
Mean no analytical solution
Median P(\mu)\,
Mode no analytical solution
Variance no analytical solution
MGF 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.


Probability density function[edit]

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).

Plot of the Logitnormal PDF for various combinations of μ (facets) and σ (colors)

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).


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


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 .

See also[edit]

Further reading[edit]

External links[edit]