he purpose of this section is to calculate distribution of time
jump of Poisson process.
Note that the
is not equal to
means that k or more jumps occurred before
means that exactly k jumps occurred before
and the union is disjoint.
We use the result (
Hence, the distribution density of the k-th arrival time is
Such distribution is called the "Gamma distribution". We will be using the
Note that by normalization we must
The integral is called "Gamma function" with the traditional
This above expression expands factorial to real and complex numbers.