he purpose of this
section is to compute some basic probabilities associated with Poisson
process. Let
be time moments,
.
The
denotes the time of the first jump. We subdivide the interval
into
small subintervals and let the size of each subinterval go to zero with
being constants. We use the result
as
and for
and the notation
,
Observe
that
We apply the
(
Chain_rule
):
From point of view of
the events
are deterministic for
,
hence,
then, by (
Poisson property
1
),
We continue in similar manner and take a
limit:
Therefore, with use of the formula
(
Poisson property 1a
), we
conclude


(Poisson property 2)

Similarly, for any integer
where
denotes any jump of
and
denotes the set of permutations of
integers from
to
.
The factorial exists because permutations of
constitute identical terms in the sum while the equality is based on
decomposition into disjoint events. We continue
We note
,
=
,
,
.
Hence,


(Poisson property 3)

