Gamma distribution

The gamma distribution is a continuous probability distribution over the set of positive real numbers. On the one hand, a direct generalization of the exponential distribution and on the other hand a generalization of the Erlang distribution for non-integer parameters. Like these, they will be used

• In queuing theory to describe the service times and repair times.
• In actuarial mathematics for modeling small to medium damage.

• 2.1 Expectation value
• 2.2 variance
• 2.3 skewness
• 2.4 reproductivity
• 2.5 Characteristic Function
• 2.6 moment generating function
• 2.7 entropy
• 2.8 Sum of gamma distributed random variables

• 3.1 Relationship to the beta distribution
• 3.2 Relationship to the chi -square distribution
• 3.3 Relationship to the Erlang distribution
• 3.4 Relationship to the exponential distribution
• 3.5 Relationship to logarithmic Gamma distribution

Definition

The pre-factor is used to correct normalization; the expression represents the function value of the gamma function, the distribution is also named.

Wherein the regularized gamma function of the upper limit.

Alternative parameterization

Alternatively to the above, common in German-speaking countries with parameterization and can also be found frequently

Is the inverse of a scale parameter and is the scale parameter itself. Density and moments change accordingly in these parameterizations ( the expectation here would be, for example, respectively). Since these parameterizations prevalent in the Anglo- Saxon world, they are most often used in the literature. To avoid misunderstanding, it is recommended that the moments explicitly specify, so to speak, for example, from a gamma distribution with mean and variance. From this parametrization and the corresponding parameter values ​​are then uniquely reconstructed.

Properties

The density has its maximum for at the point and at the points

Turning points.

Expected value

The expected value of the gamma distribution

Variance

Is the variance of the Gamma distribution

Skew

The skewness of the distribution is given by

Reproductivity

The gamma distribution is reproductively: The sum of the stochastically independent gamma distributed random variables and that both are gamma distributed with parameters and respectively, is again gamma distributed with parameters and.

Characteristic function

The characteristic feature is in the form

Moment generating function

The moment generating function of the gamma distribution is

Entropy

Is the entropy of the gamma distribution

Where ψ ( p) is the digamma function called.

Total gamma distributed random variables

Are independent gamma distributed random variables is then the sum gamma distributed, namely

In general: If is stochastically independent then

Relationship to other distributions

Relationship to the beta distribution

If and independent gamma distributed random variables with parameters and, then the size is beta distributed with parameters and, shortly

Relationship with the chi -square distribution

• The chi-square distribution with degrees of freedom is a gamma distribution with parameters and.

Relationship with the Erlang distribution

The Erlang distribution with parameters and degrees of freedom corresponds to a gamma distribution with parameters and and returns the probability of the time until the arrival of th rare, Poisson - distributed event.

Relationship to the exponential distribution

• If you choose the gamma distribution parameter, we obtain the exponential distribution with parameter.
• The convolution of n exponential distributions with the same results in a gamma distribution with.

Relationship to logarithmic Gamma distribution

Is gamma - distributed, then is distributed log - gamma.