We now set out to study the Poisson process, which
is a continuous time version of the Bernoulli process.
In the Bernoulli process, time is divided into slots.
And during each one of the slots,
we may either have an arrival or no arrival.
In the Poisson process, time is continuous.
And we may get arrivals at any time.
We want to define the Poisson process
by introducing some assumptions that in some ways parallel
the assumptions that we made for the Bernoulli process.
What where those assumptions?
The first one we made was the assumption of independence--
namely that what happens in different slots
are independent.
Similarly, for the Poisson process,
we will make the following independence assumption.
If we consider two intervals, two time intervals that
are disjoint, and look at the random variable that
stands for the number of arrivals
during this interval and that interval,
we will assume that these two random variables
are independent.
But even more than that, if we take
any collection of disjoint time intervals,
and we look at the associated random variables,
the associated numbers of arrivals,
that collection will also consist
of independent random variables.
The second assumption for the Bernoulli processes
was one of time homogeneity, namely at each slot,
we had the same probability of an arrival.
We want to make a similar assumption for the Poisson
process, and it's going to be the following.
The probability that we have k arrivals
during an interval of a certain duration tau
is going to be the same no matter where
that interval sits.
So if this is an interval that has a certain duration,
and this is an interval that has the same duration,
the probability of three arrivals in this interval
is going to be the same as the probability of three
arrivals in that interval.
And therefore, since this probability does not
depend on where the interval sits,
that probability will be fully defined
by the number of arrivals that we're interested in
and the length of the interval as
opposed to the location of the interval.
So we will be using this notation here
to indicate this probability.
In this notation, we think of tau as a constant.
And then, P of k corresponds to probabilities.
In particular, if you add over all k's
the various probabilities, what you should get
would be a value of 1, because this
exhausts all the possibilities.
And k here ranges from 0 to infinity,
because we allow an arbitrarily large number of arrivals
during a given time interval.
Now, with this assumption in place,
it would still be possible to have
arrivals that happen simultaneously,
multiple arrivals at the same time point.
In order to avoid this situation,
we introduce one more assumption which is the following.
It talks about the number of arrivals
during a time interval that has a small length delta.
During a small time interval, there
is negligible probability of having more than one arrival.
We will either have one or zero arrivals.
And the probability of one arrival
is a certain number, lambda times delta.
It's proportional to delta.
So if the interval becomes smaller and smaller,
that probability also goes to 0.
But it goes to 0 at a rate proportional to delta.
So you can think of lambda as probability per unit time.
These are the units of lambda.
Now here, I'm writing an approximate equality.
What does that mean?
It means that these are not exact expressions.
But they are exact within a second order term.
And a second order term is a term
that's negligible compared to the first order term
when delta is small.
More precisely, mathematically, what we mean
is that a second order term compared to a first order term
goes to 0 as delta goes to 0.
Finally, let me reiterate that lambda
should be interpreted as an arrival rate.
It is a probability per unit time.
The bigger lambda is, the bigger the probability
is that we get an arrival during a small time interval.
If we double lambda, then we double the probability
that we have an arrival during a small time interval.
And so we expect to have about twice as many arrivals, hence
the interpretation as an arrival rate.
We will see shortly that this is also justified
because lambda shows up in expressions for the expected
number of arrivals during a time interval.