Now that we have in our hands the PMF
of the random variable N tau, which is the number of arrivals
during an interval of length tau,
we can go ahead and try to calculate
the mean and variance of this quantity.
Regarding the mean, we could use just the definition
of the expected value and then carry out
of this calculation, which is not too hard.
And in the end, we would obtain an answer equal to lambda times
tau.
This is indeed the correct formula for the expected value.
But let us understand why this formula should
be true without doing any calculation.
Remember that the random variable, the number
of arrivals in the Poisson process,
is well approximated by a binomial random variable
with those particular parameters n and p in the limit
when delta goes to zero.
And this works through a discretization argument.
Therefore, the expected value of N tau
should be approximately equal to the expected value of that we
get from the Bernoulli processes, that is the expected
value of the binomial random variable.
And the expected value of a binomial random variable
is n times p.
And n times p evaluates approximately to lambda times
tau.
The second equality here is approximate because we're
neglecting this order of delta squared term.
Now, these approximations are increasingly
exact as we let delta go to 0.
And when we take the limit as delta goes to 0,
we see that the expected value of the number of arrivals
in the Poisson process will be equal to lambda tau.
For the variance, we can follow a similar argument.
First do a binomial approximation
and use the formula for the variance
of a binomial random variable.
And then, when delta is small, this number p is small.
And it's negligible compared to 1.
n times p is approximately lambda [tau].
And so we obtain this expression
This expression here is the correct one.
If we were to use the formal definition of the variance
and carry out the calculations using the PMF,
this is what we would obtain, except that it
would be somewhat tedious.
The formulas that we have derived,
at least the first one, is quite intuitive
and has a natural form.
The expected number of arrivals is proportional to tau.
If we double the length of the time interval for interest,
we expect to see twice as many arrivals.
This formula also reinforces the interpretation of lambda
as an arrival rate.
Since lambda is the expected number of arrivals divided
by the length of time, it is, really,
the expected number of arrivals per unit time.
So it's natural to call lambda the arrival rate,
or the intensity of the arrival process.
Finally, regarding the variance, it is a curious fact
that the variance turns out to be
exactly the same as the expected value.
And this is a special property of the Poisson PMF.