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 definitionWe 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?)