he purpose of this section is to calculate a distribution of the time
of exactly
-th
jump of the Poisson process.
Note that the
Prob
is not equal to
Prob
.
Indeed, the
means that k or more jumps occurred. The
means that exactly k jumps occurred.
and the union is disjoint.
Hence,
We use the result (
Poisson property
3
):
Consequently,
Hence, the distribution density of the k-th arrival time is
Such distribution is called the "Gamma distribution". Following the Wikipedia
we will be using the following notation
Note, that by normalisation we should
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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