Random Variable Distribution Types

Gamma Distribution

The gamma distribution ("Figure: Input Variable Wizard - Gamma Distribution") is a two parameter, flexible, continuous probability distribution. The two parameters are Shape and Scale. This distribution can be either symmetrical or positively skewed.

Triangular Distribution

The triangular distribution ("Figure: Input Variable Wizard - Triangular Distribution") is a three-parameter continuous probability distribution. The three parameters are Left (minimum value), Right (maximum value) and Peak (the mode, or most-likely value, where the peak of the triangle is located). This distribution is useful in that it can be symmetrical, or it can be asymmetrical with either a positive or negative skew, i.e., a longer upper or longer lower tail. For example, a negative skew allows the value to be farther below the most-likely value than it can be above.

Empirical Distribution

An empirical (graphical) distribution ("Figure: Input Variable Wizard - Empirical Distribution") is continuous but without a function, for which the likelihood of possible values is derived from observed data. The empirical distribution is represented by a lookup table that describes points on the cumulative distribution function (CDF).

Uniform Distribution

A uniform distribution ("Figure: Input Variable Wizard - Uniform Distribution") is a distribution that has a constant probability across the possible range, with any value having an equal likelihood or probability. The parameters for this distribution are Left (minimum value) and Right (maximum value), defining the possible range.

Discrete Distribution

A discrete distribution ("Figure: Input Variable Wizard - Discrete Distribution") describes the probability of occurrence of each possible value of a discrete random variable. A discrete random variable is a random variable that has a limited set of possible values, such as the values on the faces of a six-sided die. A discrete distribution is defined by a table of individual values and their probabilities.

If your input variable is a time series, such as reservoir inflow, the only distribution type available is Discrete, since sampling for an input time series is done only from a set of time series records that you identify. "Figure: Discrete Distribution for Input Time Series" shows how the parameter table for a discrete distribution changes for an input time series variable, as opposed to other variable types. Weight is a relative value, and probability is then computed as weight of each divided by the sum of weights.

Normal Distribution

A normal distribution ("Figure: Input Variable Wizard - Normal Distribution") is a continuous, two parameter, symmetrical probability distribution. The two parameters are mean and standard deviation.

LogNormal Distribution

A log-normal (or lognormal) distribution ("Figure: Input Variable Wizard - LogNormal Distribution") is a continuous probability distribution of a random variable whose (natural) logarithm is normally distributed. The parameters are the mean of the logarithm of the variable, and the standard deviation of the logarithm of the variable. It is always positively skewed, by a degree dependent on the ratio of mean to standard deviation.

Log10Normal Distribution

A log10-normal (or log10normal) distribution ("Figure: Input Variable Wizard - Log10Normal Distribution") is a continuous probability distribution of a random variable whose base10 logarithm is normally distributed. The parameters are the mean of the logarithm of the variable, and the standard deviation of the logarithm of the variable. It is always positively skewed, by a degree dependent on the ratio of mean to standard deviation.