he purpose of this section is to calculate distribution of time
of exactly
-th
jump of Poisson process.
Note that the
Prob
is not equal to
Prob
.
Indeed, the
means that k or more jumps occurred before
.
The event
means that exactly k jumps occurred before
.
and the union is disjoint.
Hence,
We use the result (
Poisson property
3
):
Therefore
Hence, the distribution density of the k-th arrival time is
Such distribution is called the "Gamma distribution". We will be using the
following notation
Note that by normalization we must
have
hence
The integral is called "Gamma function" with the traditional
notation
This above expression expands factorial to real and complex numbers.
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