# Gamma Distribution

The **Gamma** distribution is widely used in engineering, science, and business, to model continuous variables that are always positive and have skewed distributions. In **CPillar**, the Gamma distribution can be useful for any variable which is always positive, such as cohesion or shear strength for example.

The Gamma distribution has the following probability density function:

where G(a) is the Gamma function, and the parameters a and b are both positive, i.e. a > 0 and b > 0.

- a (alpha) is known as the shape parameter, while b (beta) is referred to as the scale parameter.
- b has the effect of stretching or compressing the range of the Gamma distribution. A Gamma distribution with b = 1 is known as the standard Gamma distribution.

The Gamma distribution represents a family of shapes. As suggested by its name, a controls the shape of the family of distributions. The fundamental shapes are characterized by the following values of a:

Case I (a < 1)

When a < 1, the Gamma distribution is exponentially shaped and asymptotic to both the vertical and horizontal axes.

Case II (a = 1)

A Gamma distribution with shape parameter a = 1 and scale parameter b is the same as an exponential distribution of scale parameter (or mean) b.

Case III (a > 1)

When a is greater than one, the Gamma distribution assumes a mounded (unimodal), but skewed shape. The skewness reduces as the value of a increases.

Examples of shapes of the standard Gamma distributions with different values of a are shown in the figure below.

The shape and scale parameters of a Gamma distribution can be calculated from its mean m and standard deviation *s* according to the relationships:

From the expression for a , it can be seen that:

- Case I of the Gamma distribution’s shape occurs when the mean m is less than the standard deviation
*s*. - Case II – the case of the exponential distribution – occurs when the mean is equal to the standard deviation.
- The third shape of the Gamma distribution arises when the mean is greater than the standard deviation.

The Gamma distribution is sometimes called the Erlang distribution, when its shape parameter a is an integer.