We have seen that the binomial distribution plays
an important role in the study of the Bernoulli process.
And the reason is that the binomial distribution describes
the number of arrivals during a fixed number of slots.
We will now develop an approximation
to the binomial distribution that
applies to one particular regime,
and that regime is when we have a very large number of slots,
but we have a small probability of success in each slot.
And this is in a way so that the product of these two numbers,
which is the expected number of successes,
is a moderate number.
One example where such a situation might arise
is the following.
Suppose you're interested in earthquakes in your city,
and you divide time into slots of one hour.
During each hour, the probability
of having a noticeable earthquake in your city
would be a very small number.
On the other hand, if you're interested in a time
frame of five years, there's going
to be many hours during that time frame,
so that n would be quite large.
But the expected number of earthquakes
over a period of five years should be a moderate number.
And one can think of other situations
where this regime might arise.
The one particular situation that
will be very interesting for us is
going to be when we try to take a continuous time
approximation of the Bernoulli process
by dividing time into very small slots,
so that we have many slots, but a small probability of success
during each one of those slots.
So to start, let us look at the form of the binomial PMF.
And let us just try to develop an approximation to this PMF,
when we fix k to be particular constant number,
and then take the limit as n goes to infinity and p
goes to 0, but in a way that lambda remains constant.
And in particular, because of this relation
here, we will have p equal to lambda over n.
So let us take this expression and start rewriting it.
Let us look at the ratio of n factorial divided by this.
The denominator has the product of all numbers going up
to n minus k.
So by dividing by this number, what is left out of the n
factorial is only the terms that go up to n minus k plus 1.
Then we have, in the denominator, the factor
of k factorial.
Now p is equal to lambda over n, so this term becomes lambda
to the k divided by n to the k.
And similarly, for the last term,
we have 1 minus lambda over n to the power n minus k.
Now let us rearrange terms.
Here, we have a product of k terms in the numerator.
Here, we have n multiplying itself k times.
So we can take a factor of n and place it
underneath each one of those terms
to obtain n over n times n minus 1 over n times--
we continue this way all the way to n minus k plus 1 divided
by n.
Take this term, k factorial, move it
underneath the lambda to the k term,
and then let us split this last term
into 2 pieces in this manner.
And now let us start taking limits as n goes to infinity.
The first term that we have here is equal to 1.
How about the second term?
n divided by n is equal to 1, 1 over n goes to 0,
so this term also converges to 1.
And by a similar argument, all of the terms in this product,
including this one, converge to 1.
The term lambda k over k factorial
remains exactly as is.
And now, let us look at this term.
This is probably familiar.
There is a basic fact which tells us
that if we take this expression and raise it
to the nth power, what we get is e to the minus
lambda in the limit as n goes to infinity.
So using this basic result, this term
becomes e to the minus lambda.
And finally, let's look at the last term.
Remember that k is fixed, is a constant.
1 minus lambda over n converges to 1,
and when we raise that number to the k-th power,
we still get a 1 in the limit.
So the only terms that are left are here,
and essentially, what we have just
established is that in the limit,
the probability of k arrivals in a Bernoulli process
or the binomial probability evaluated
at k, in the limit, as n goes to infinity and p goes to 0,
is given by this formula, here.
This is the formula for the Poisson PMF.
And so what we have established is that the binomial PMF
converges to a Poisson PMF when we
take the limit in this particular way.