This density is called the density. In general the gamma density is defined with 2 parameters (both positive reals, most often

where is the constant that makes the integral of the density sum to one:

By integration by parts we showed the important recurrence relation:

Because , we have for integer

The particular case of the integer *t* can be compared to the sum of
*n* independent exponentials, it is the waiting time to the *nth*
event,
it is the *twin* of the negative binomial.

From this we can guess what the expected value and the variance are
going to be:
If all the *X*_{i}'s are independent
,
then if we sum *n* of them we have
and if they are independent:

This generalizes to the non integer *t* case: