|Probability density function
|Cumulative distribution function
|Parameters||σ2 > 0 — squared scale (real),
μ ∈ R — location
|Support||x ∈ (0, 1)|
|Mean||no analytical solution|
|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, which often refers to a multinomial logit version (e.g.    ).
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
The probability density function of a logit-normal distribution is:
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:
- Beta distribution and Kumaraswamy distribution, other two-parameter distributions on a bounded interval with similar shapes
- Frederic, P. & Lad, F. (2008) Two Moments of the Logitnormal Distribution. Communications in Statistics-Simulation and Computation. 37: 1263-1269
- Mead, R. (1965). "A Generalised Logit-Normal Distribution". Biometrics 21 (3): 721–732. doi:10.2307/2528553. JSTOR 2528553.
- J Atchison and SM Shen. "Logistic-normal distributions: Some properties and uses." Biometrika, 1980. Google Scholar link
- Peter Hoff, 2003. Link