# Continuous Statistical Distributions¶

## Overview¶

All distributions will have location (L) and Scale (S) parameters along with any shape parameters needed, the names for the shape parameters will vary. Standard form for the distributions will be given where and The nonstandard forms can be obtained for the various functions using (note is a standard uniform random variate).

Function Name Standard Function Transformation
Cumulative Distribution Function (CDF)  Probability Density Function (PDF)  Percent Point Function (PPF)  Probability Sparsity Function (PSF)  Hazard Function (HF)  Cumulative Hazard Functon (CHF)   Survival Function (SF)  Inverse Survival Function (ISF)  Moment Generating Function (MGF)  Random Variates  (Differential) Entropy  (Non-central) Moments  Central Moments  mean (mode, median), var  skewness, kurtosis   ### Moments¶

Non-central moments are defined using the PDF Note, that these can always be computed using the PPF. Substitute in the above equation and get which may be easier to compute numerically. Note that so that Central moments are computed similarly  In particular Skewness is defined as while (Fisher) kurtosis is so that a normal distribution has a kurtosis of zero.

### Median and mode¶

The median, is defined as the point at which half of the density is on one side and half on the other. In other words, so that In addition, the mode, , is defined as the value for which the probability density function reaches it’s peak ### Fitting data¶

To fit data to a distribution, maximizing the likelihood function is common. Alternatively, some distributions have well-known minimum variance unbiased estimators. These will be chosen by default, but the likelihood function will always be available for minimizing.

If is the PDF of a random-variable where is a vector of parameters ( e.g. and ), then for a collection of independent samples from this distribution, the joint distribution the random vector is The maximum likelihood estimate of the parameters are the parameters which maximize this function with fixed and given by the data: Where Note that if includes only shape parameters, the location and scale-parameters can be fit by replacing with in the log-likelihood function adding and minimizing, thus If desired, sample estimates for and (not necessarily maximum likelihood estimates) can be obtained from samples estimates of the mean and variance using where and are assumed known as the mean and variance of the untransformed distribution (when and ) and ### Standard notation for mean¶

We will use where should be clear from context as the number of samples 