GAMMA DISTRIBUTION ASSIGNMENT HELP

The gamma function is given by the formula below;

Where α is a parameter popularly referred to as the shape parameter. It is this parameter that influences the shape of the gamma distribution if you were to plot it.
When α = 1 we obtain the following result from the gamma function;

When x = λy we obtain the following result from the gamma function;

Properties

For all values of the shape parameter greater than 0, i.e. α> 0, the following properties hold;
Γ(α+1)=αΓ(α)
This result is obtained using integration by parts.
Γ(n+1)=n!
This is true for all values of n contained in the set of positive integers.
Γ(1/2)=√π
The gamma distribution of a random variable X with parameters α and β is given by the following probability density function (pdf);

Where β is a parameter popularly referred to as the scale parameter. M_x (t)=1/(1-βt)^α

Statistical Measures

Below is a quick way of computing the mean and variance of a random variable X that has gamma distribution;
Mean = αβ
Variance = αβ²
When β = 1 we obtain the following result from the gamma distribution;

This new distribution is referred to as the standard gamma distribution.

The incomplete gamma function of a random variable X that has gamma distribution is given by the following cumulative distribution function (cdf);