HW Math 504. Spring 2008. CSUF
Problems 9.3 and 9.5
by Nasser Abbasi
Start by showing that the processes
and
are each a Poisson process. Next show that they are independent by showing
that the product of these 2 distributions is equal to the joined distribution.
Given:
,
Where are told that
is a Poisson process. Need to find
and
.
By law of total probabilities
Hence
Similarly,
Now find expression for the joined distribution
to complete the above evaluation. Condition on
hence we obtain
or
But since
,
then the above reduces to one case which is
And all the other probabilities must be zero.
Now in (1), we are given that
is a Poisson process with some rate
Hence the rate adjusted for duration
must be
,
hence from definition of Poisson process with rate
we
write
Now we need to evaluate the term
in (1). This terms asks for the probability of getting the sum
.
If we think of
as number of successes and
as number of failures,
Where
is the probability of event type
,
and
is the probability of not getting this event, which is the probability of
event
which
is given by
Substitute (2) and (3) into (1) we obtain
But
,
hence
hence (4) becomes
The above is the joined probability of
and
We know can determine the probability distribution of
and
from substituting (5) into (A1) and (A2)
We remove terms outside sum which do not depend on
and obtain
But
by definition, hence the above becomes
Therefore, we see that
satisfies the Poisson formula. To show it is a Poisson distribution, we must
also show that it satisfies the following
We
see that at
,
the above becomes
But
,
hence
Increments are independents of each others. Since the original process
is already given to be Poisson process, then the increments of
are independent of each others. But
increments are a subset of those increments. Therefore,
increments must by necessity be independent of each others.
Similar arguments show that
and that it satisfies the Poisson
definition
We
now need to show independence. We see that
But from (5) above, we see this is the same as
,
therefore
Hence
and
are 2 independent Poisson processes.
Let the interr arival time between each car be
where
is the interval as indicated by this diagram
Inter-arrival times are
random variable which is exponential distributed
For the number of cars that pass through the intersection to be
it must imply that the interval between the first
cars was less than
and that the
car arrived after than
car after more than
units of time. Therefore
But since
is a Poisson random number with parameter
, then the time between increment
is an exponential random number with parameter
(and they are independent from each others). Hence
and
Hence (1) becomes
This is a small program which plots the probability above as function of
for some fixed
.
It shows as expected the probability of
becomes smaller the larger
gets.
Now
Let
then the above becomes
The above sum converges since by ratio test the
term over the
term is less than one. (I can find a closed form expression for this sum?)