WEBVTT

00:00:00.560 --> 00:00:02.950
We have seen that the
binomial distribution plays

00:00:02.950 --> 00:00:06.020
an important role in the study
of the Bernoulli process.

00:00:06.020 --> 00:00:10.440
And the reason is that the
binomial distribution describes

00:00:10.440 --> 00:00:16.500
the number of arrivals during
a fixed number of slots.

00:00:16.500 --> 00:00:18.770
We will now develop
an approximation

00:00:18.770 --> 00:00:20.870
to the binomial
distribution that

00:00:20.870 --> 00:00:23.650
applies to one
particular regime,

00:00:23.650 --> 00:00:29.250
and that regime is when we have
a very large number of slots,

00:00:29.250 --> 00:00:33.720
but we have a small probability
of success in each slot.

00:00:33.720 --> 00:00:38.710
And this is in a way so that the
product of these two numbers,

00:00:38.710 --> 00:00:41.700
which is the expected
number of successes,

00:00:41.700 --> 00:00:44.500
is a moderate number.

00:00:44.500 --> 00:00:47.500
One example where such
a situation might arise

00:00:47.500 --> 00:00:48.400
is the following.

00:00:48.400 --> 00:00:51.550
Suppose you're interested
in earthquakes in your city,

00:00:51.550 --> 00:00:55.330
and you divide time
into slots of one hour.

00:00:55.330 --> 00:00:57.700
During each hour,
the probability

00:00:57.700 --> 00:01:00.640
of having a noticeable
earthquake in your city

00:01:00.640 --> 00:01:02.790
would be a very small number.

00:01:02.790 --> 00:01:04.900
On the other hand, if
you're interested in a time

00:01:04.900 --> 00:01:07.430
frame of five
years, there's going

00:01:07.430 --> 00:01:10.470
to be many hours
during that time frame,

00:01:10.470 --> 00:01:13.260
so that n would be quite large.

00:01:13.260 --> 00:01:15.680
But the expected
number of earthquakes

00:01:15.680 --> 00:01:20.620
over a period of five years
should be a moderate number.

00:01:20.620 --> 00:01:22.660
And one can think
of other situations

00:01:22.660 --> 00:01:25.200
where this regime might arise.

00:01:25.200 --> 00:01:26.910
The one particular
situation that

00:01:26.910 --> 00:01:29.600
will be very
interesting for us is

00:01:29.600 --> 00:01:33.030
going to be when we try
to take a continuous time

00:01:33.030 --> 00:01:35.310
approximation of the
Bernoulli process

00:01:35.310 --> 00:01:38.479
by dividing time into
very small slots,

00:01:38.479 --> 00:01:42.500
so that we have many slots, but
a small probability of success

00:01:42.500 --> 00:01:44.750
during each one of those slots.

00:01:44.750 --> 00:01:48.990
So to start, let us look at
the form of the binomial PMF.

00:01:48.990 --> 00:01:53.320
And let us just try to develop
an approximation to this PMF,

00:01:53.320 --> 00:01:58.729
when we fix k to be
particular constant number,

00:01:58.729 --> 00:02:06.160
and then take the limit as
n goes to infinity and p

00:02:06.160 --> 00:02:11.970
goes to 0, but in a way that
lambda remains constant.

00:02:11.970 --> 00:02:15.320
And in particular,
because of this relation

00:02:15.320 --> 00:02:21.490
here, we will have p
equal to lambda over n.

00:02:21.490 --> 00:02:26.870
So let us take this expression
and start rewriting it.

00:02:26.870 --> 00:02:31.260
Let us look at the ratio of
n factorial divided by this.

00:02:31.260 --> 00:02:34.720
The denominator has the
product of all numbers going up

00:02:34.720 --> 00:02:36.420
to n minus k.

00:02:36.420 --> 00:02:41.070
So by dividing by this number,
what is left out of the n

00:02:41.070 --> 00:02:48.790
factorial is only the terms
that go up to n minus k plus 1.

00:02:53.170 --> 00:02:56.140
Then we have, in the
denominator, the factor

00:02:56.140 --> 00:02:58.190
of k factorial.

00:02:58.190 --> 00:03:03.740
Now p is equal to lambda over
n, so this term becomes lambda

00:03:03.740 --> 00:03:07.480
to the k divided by n to the k.

00:03:07.480 --> 00:03:09.640
And similarly,
for the last term,

00:03:09.640 --> 00:03:16.650
we have 1 minus lambda over
n to the power n minus k.

00:03:19.160 --> 00:03:22.450
Now let us rearrange terms.

00:03:22.450 --> 00:03:26.550
Here, we have a product of
k terms in the numerator.

00:03:26.550 --> 00:03:29.720
Here, we have n
multiplying itself k times.

00:03:29.720 --> 00:03:33.130
So we can take a factor
of n and place it

00:03:33.130 --> 00:03:35.300
underneath each
one of those terms

00:03:35.300 --> 00:03:42.180
to obtain n over n times
n minus 1 over n times--

00:03:42.180 --> 00:03:46.680
we continue this way all the
way to n minus k plus 1 divided

00:03:46.680 --> 00:03:48.280
by n.

00:03:48.280 --> 00:03:50.970
Take this term, k
factorial, move it

00:03:50.970 --> 00:03:54.900
underneath the
lambda to the k term,

00:03:54.900 --> 00:03:57.570
and then let us
split this last term

00:03:57.570 --> 00:04:06.010
into 2 pieces in this manner.

00:04:06.010 --> 00:04:10.915
And now let us start taking
limits as n goes to infinity.

00:04:15.680 --> 00:04:19.610
The first term that we
have here is equal to 1.

00:04:19.610 --> 00:04:21.620
How about the second term?

00:04:21.620 --> 00:04:25.690
n divided by n is equal
to 1, 1 over n goes to 0,

00:04:25.690 --> 00:04:28.940
so this term also
converges to 1.

00:04:28.940 --> 00:04:33.159
And by a similar argument, all
of the terms in this product,

00:04:33.159 --> 00:04:36.990
including this
one, converge to 1.

00:04:36.990 --> 00:04:39.900
The term lambda k
over k factorial

00:04:39.900 --> 00:04:42.530
remains exactly as is.

00:04:42.530 --> 00:04:45.450
And now, let us
look at this term.

00:04:45.450 --> 00:04:47.770
This is probably familiar.

00:04:47.770 --> 00:04:51.150
There is a basic
fact which tells us

00:04:51.150 --> 00:04:53.570
that if we take this
expression and raise it

00:04:53.570 --> 00:04:57.500
to the nth power, what
we get is e to the minus

00:04:57.500 --> 00:05:01.390
lambda in the limit
as n goes to infinity.

00:05:01.390 --> 00:05:03.610
So using this basic
result, this term

00:05:03.610 --> 00:05:06.010
becomes e to the minus lambda.

00:05:06.010 --> 00:05:09.320
And finally, let's
look at the last term.

00:05:09.320 --> 00:05:14.490
Remember that k is
fixed, is a constant.

00:05:14.490 --> 00:05:18.822
1 minus lambda over
n converges to 1,

00:05:18.822 --> 00:05:22.570
and when we raise that
number to the k-th power,

00:05:22.570 --> 00:05:27.200
we still get a 1 in the limit.

00:05:27.200 --> 00:05:31.560
So the only terms that
are left are here,

00:05:31.560 --> 00:05:33.490
and essentially,
what we have just

00:05:33.490 --> 00:05:36.330
established is
that in the limit,

00:05:36.330 --> 00:05:42.159
the probability of k arrivals
in a Bernoulli process

00:05:42.159 --> 00:05:44.890
or the binomial
probability evaluated

00:05:44.890 --> 00:05:51.000
at k, in the limit, as n goes
to infinity and p goes to 0,

00:05:51.000 --> 00:05:54.850
is given by this formula, here.

00:05:54.850 --> 00:05:56.620
This is the formula
for the Poisson PMF.

00:05:59.340 --> 00:06:04.400
And so what we have established
is that the binomial PMF

00:06:04.400 --> 00:06:08.280
converges to a
Poisson PMF when we

00:06:08.280 --> 00:06:12.050
take the limit in
this particular way.