®
The Poisson
Distribution 37.3
Introduction
In this Section we introduce a probability model which can be used when the outcome of an experiment
is a random variable taking on positive integer values and where the only information available is a
measurement of its average value. This has widespread applications, for example in analysing traffic
flow, in fault prediction on electric cables and in the prediction of randomly occurring accidents. We
shall look at the Poisson distribution in two distinct ways. Firstly, as a distribution in its own right.
This will enable us to apply statistical methods to a set of problems which cannot be solved using
the binomial distribution. Secondly, as an approximation to the binomial distribution X ∼ B(n, p)
in the case where n is large and p is small. You will find that this approximation can often save the
need to do much tedious arithmetic.
• understand the concepts of probability
Prerequisites
• understand the concepts and notation for the
Before starting this Section you should . . . binomial distribution
'
$
• recognise and use the formula for probabilities
calculated from the Poisson model
Learning Outcomes • use the recurrence relation to generate a
succession of probabilities
On completion you should be able to . . .
• use the Poisson model to obtain approximate
values for binomial probabilities
& %
HELM (2008): 37
Section 37.3: The Poisson Distribution
,1. The Poisson approximation to the binomial distribution
The probability of the outcome X = r of a set of Bernoulli trials can always be calculated by using
the formula
n
P(X = r) = Cr q n−r pr
given above. Clearly, for very large values of n the calculation can be rather tedious, this is particularly
so when very small values of p are also present. In the situation when n is large and p is small and the
product np is constant we can take a different approach to the problem of calculating the probability
that X = r. In the table below the values of P(X = r) have been calculated for various combinations
of n and p under the constraint that np = 1. You should try some of the calculations for yourself
using the formula given above for some of the smaller values of n.
Probability of X successes
n p X=0 X=1 X=2 X=3 X=4 X=5 X=6
4 0.25 0.316 0.422 0.211 0.047 0.004
5 0.20 0.328 0.410 0.205 0.051 0.006 0.000
10 0.10 0.349 0.387 0.194 0.058 0.011 0.001 0.000
20 0.05 0.359 0.377 0.189 0.060 0.013 0.002 0.000
100 0.01 0.366 0.370 0.185 0.061 0.014 0.003 0.001
1000 0.001 0.368 0.368 0.184 0.061 0.015 0.003 0.001
10000 0.0001 0.368 0.368 0.184 0.061 0.015 0.003 0.001
Each of the binomial distributions given has a mean given by np = 1. Notice that the probabilities
that X = 0, 1, 2, 3, 4, . . . approach the values 0.368, 0.368, 0.184, . . . as n increases.
If we have to determine the probabilities of success when large values of n and small values of p are
involved it would be very convenient if we could do so without having to construct tables. In fact we
can do such calculations by using the Poisson distribution which, under certain constraints, may be
considered as an approximation to the binomial distribution.
By considering simplifications applied to the binomial distribution subject to the conditions
1. n is large
2. p is small
3. np = λ (λ a constant)
we can derive the formula
λr
P(X = r) = e−λ as an approximation to P(X = r) = n Cr q n−r pr .
r!
This is the Poisson distribution given previously. We now show how this is done. We know that the
binomial distribution is given by
n(n − 1) n−2 2 n(n − 1) . . . (n − r + 1) n−r r
(q + p)n = q n + nq n−1 p + q p + ··· + q p + · · · + pn
2! r!
Condition (2) tells us that since p is small, q = 1 − p is approximately equal to 1. Applying this to
the terms of the binomial expansion above we see that the right-hand side becomes
n(n − 1) 2 n(n − 1) . . . (n − r + 1) r
1 + np + p + ··· + p + · · · + pn
2! r!
38 HELM (2008):
Workbook 37: Discrete Probability Distributions
, ®
Applying condition (1) allows us to approximate terms such as (n − 1), (n − 2), . . . to n (mathemat-
ically, we are allowing n → ∞ ) and the right-hand side of our expansion becomes
n2 2 nr r
1 + np + p + · · · + p + . . .
2! r!
n
Note that the term p → 0 under these conditions and hence has been omitted.
We now have the series
(np)2 (np)r
1 + np + + ··· + + ...
2! r!
which, using condition (3) may be written as
(λ)2 (λ)r
1+λ+ + ··· + + ...
2! r!
You may recognise this as the expansion of eλ .
If we are to be able to claim that the terms of this expansion represent probabilities, we must be sure
that the sum of the terms is 1. We divide by eλ to satisfy this condition. This gives the result
eλ 1 (λ)2 (λ)r
= 1 = (1 + λ + + · · · + + ...)
eλ eλ 2! r!
λ2 λ3 λr
= e−λ + e−λ λ + e−λ + e−λ + · · · + e−λ + · · · +
2! 3! r!
The terms of this expansion are very good approximations to the corresponding binomial expansion
under the conditions
1. n is large
2. p is small
3. np = λ (λ constant)
The Poisson approximation to the binomial distribution is summarized below.
Key Point 6
Poisson Approximation to the Binomial Distribution
Assuming that n is large, p is small and that np is constant, the terms
n
P(X = r) = Cr (1 − p)n−r pr
of a binomial distribution may be closely approximated by the terms
r
−λ λ
P(X = r) = e
r!
of the Poisson distribution for corresponding values of r.
HELM (2008): 39
Section 37.3: The Poisson Distribution
The Poisson
Distribution 37.3
Introduction
In this Section we introduce a probability model which can be used when the outcome of an experiment
is a random variable taking on positive integer values and where the only information available is a
measurement of its average value. This has widespread applications, for example in analysing traffic
flow, in fault prediction on electric cables and in the prediction of randomly occurring accidents. We
shall look at the Poisson distribution in two distinct ways. Firstly, as a distribution in its own right.
This will enable us to apply statistical methods to a set of problems which cannot be solved using
the binomial distribution. Secondly, as an approximation to the binomial distribution X ∼ B(n, p)
in the case where n is large and p is small. You will find that this approximation can often save the
need to do much tedious arithmetic.
• understand the concepts of probability
Prerequisites
• understand the concepts and notation for the
Before starting this Section you should . . . binomial distribution
'
$
• recognise and use the formula for probabilities
calculated from the Poisson model
Learning Outcomes • use the recurrence relation to generate a
succession of probabilities
On completion you should be able to . . .
• use the Poisson model to obtain approximate
values for binomial probabilities
& %
HELM (2008): 37
Section 37.3: The Poisson Distribution
,1. The Poisson approximation to the binomial distribution
The probability of the outcome X = r of a set of Bernoulli trials can always be calculated by using
the formula
n
P(X = r) = Cr q n−r pr
given above. Clearly, for very large values of n the calculation can be rather tedious, this is particularly
so when very small values of p are also present. In the situation when n is large and p is small and the
product np is constant we can take a different approach to the problem of calculating the probability
that X = r. In the table below the values of P(X = r) have been calculated for various combinations
of n and p under the constraint that np = 1. You should try some of the calculations for yourself
using the formula given above for some of the smaller values of n.
Probability of X successes
n p X=0 X=1 X=2 X=3 X=4 X=5 X=6
4 0.25 0.316 0.422 0.211 0.047 0.004
5 0.20 0.328 0.410 0.205 0.051 0.006 0.000
10 0.10 0.349 0.387 0.194 0.058 0.011 0.001 0.000
20 0.05 0.359 0.377 0.189 0.060 0.013 0.002 0.000
100 0.01 0.366 0.370 0.185 0.061 0.014 0.003 0.001
1000 0.001 0.368 0.368 0.184 0.061 0.015 0.003 0.001
10000 0.0001 0.368 0.368 0.184 0.061 0.015 0.003 0.001
Each of the binomial distributions given has a mean given by np = 1. Notice that the probabilities
that X = 0, 1, 2, 3, 4, . . . approach the values 0.368, 0.368, 0.184, . . . as n increases.
If we have to determine the probabilities of success when large values of n and small values of p are
involved it would be very convenient if we could do so without having to construct tables. In fact we
can do such calculations by using the Poisson distribution which, under certain constraints, may be
considered as an approximation to the binomial distribution.
By considering simplifications applied to the binomial distribution subject to the conditions
1. n is large
2. p is small
3. np = λ (λ a constant)
we can derive the formula
λr
P(X = r) = e−λ as an approximation to P(X = r) = n Cr q n−r pr .
r!
This is the Poisson distribution given previously. We now show how this is done. We know that the
binomial distribution is given by
n(n − 1) n−2 2 n(n − 1) . . . (n − r + 1) n−r r
(q + p)n = q n + nq n−1 p + q p + ··· + q p + · · · + pn
2! r!
Condition (2) tells us that since p is small, q = 1 − p is approximately equal to 1. Applying this to
the terms of the binomial expansion above we see that the right-hand side becomes
n(n − 1) 2 n(n − 1) . . . (n − r + 1) r
1 + np + p + ··· + p + · · · + pn
2! r!
38 HELM (2008):
Workbook 37: Discrete Probability Distributions
, ®
Applying condition (1) allows us to approximate terms such as (n − 1), (n − 2), . . . to n (mathemat-
ically, we are allowing n → ∞ ) and the right-hand side of our expansion becomes
n2 2 nr r
1 + np + p + · · · + p + . . .
2! r!
n
Note that the term p → 0 under these conditions and hence has been omitted.
We now have the series
(np)2 (np)r
1 + np + + ··· + + ...
2! r!
which, using condition (3) may be written as
(λ)2 (λ)r
1+λ+ + ··· + + ...
2! r!
You may recognise this as the expansion of eλ .
If we are to be able to claim that the terms of this expansion represent probabilities, we must be sure
that the sum of the terms is 1. We divide by eλ to satisfy this condition. This gives the result
eλ 1 (λ)2 (λ)r
= 1 = (1 + λ + + · · · + + ...)
eλ eλ 2! r!
λ2 λ3 λr
= e−λ + e−λ λ + e−λ + e−λ + · · · + e−λ + · · · +
2! 3! r!
The terms of this expansion are very good approximations to the corresponding binomial expansion
under the conditions
1. n is large
2. p is small
3. np = λ (λ constant)
The Poisson approximation to the binomial distribution is summarized below.
Key Point 6
Poisson Approximation to the Binomial Distribution
Assuming that n is large, p is small and that np is constant, the terms
n
P(X = r) = Cr (1 − p)n−r pr
of a binomial distribution may be closely approximated by the terms
r
−λ λ
P(X = r) = e
r!
of the Poisson distribution for corresponding values of r.
HELM (2008): 39
Section 37.3: The Poisson Distribution
