## Gamma distribution

**Definition**. The **gamma distribution** is a two-parametric family of densities. For the density is defined by

Obviously, you need to know what is a gamma function. My notation of the parameters follows Feller, W. An Introduction to Probability Theory and its Applications, Volume II, 2nd edition (1971). It is different from the one used by J. Abdey in his guide ST2133.

### Property 1

It is really a density because

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Suppose you see an expression and need to determine which gamma density this is. The power of the exponent gives you and the power of gives you It follows that the normalizing constant should be and the density is

### Property 2

The most important property is that the family of gamma densities with the same is closed under convolutions. Because of the associativity property it is enough to prove this for the case of two gamma densities.

First we want to prove

(1)

Start with the general definition of convolution and recall where the density vanishes:

(plug the densities and take out the constants)

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Thus (1) is true. Integrating it we have

We know that the convolution of two densities is a density. Therefore the last equation implies

and

**Alternative proof**. The moment generating function of a sum of two independent beta distributions with the same shows that this sum is again a beta distribution with the same , see pp. 141, 209 in the guide ST2133.