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In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable X is log-normally distributed, then Y = ln( X ) has a normal distribution.
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 t is the standard logistic function, then X = t(Y) has a logit-normal distribution; likewise, if X is logit-normally distributed, then Y = logit(X)= log (X/(1-X)) is normally distributed.
Such variables may be better described by other distributions, such as the log-normal distribution or the Pareto distribution. The value of the normal density is practically zero when the value x {\displaystyle x} lies more than a few standard deviations away from the mean (e.g., a spread of three standard deviations covers all but 0.27 ...
Where ( ) is the inverse standardized Student t CDF, and ( ) is the standardized Student t PDF. [ 2 ] In probability theory and statistics , Student's t distribution (or simply the t distribution ) t ν {\displaystyle \ t_{\nu }\ } is a continuous probability distribution that generalizes the standard normal distribution .
Euler Mathematical Toolbox (or EuMathT; formerly Euler) is a free and open-source numerical software package. It contains a matrix language, a graphical notebook style interface, and a plot window. Euler is designed for higher level math such as calculus, optimization, and statistics.
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Another generalized log-logistic distribution is the log-transform of the metalog distribution, in which power series expansions in terms of are substituted for logistic distribution parameters and . The resulting log-metalog distribution is highly shape flexible, has simple closed form PDF and quantile function , can be fit to data with linear ...
The metalog distribution is a generalization of the logistic distribution, where the term "metalog" is short for "metalogistic".Starting with the logistic quantile function, = = + (), Keelin substituted power series expansions in cumulative probability = for the and the parameters, which control location and scale, respectively.