Download A Normal Approximation to the Poisson Distribution

A Normal Approximation to the
Poisson Distribution
NORMAL APPROXIMATION TO POISSON
The following sketches illustrate the distributions of a
Poisson distribution X with different mean (α) values.
X ~ Po (α)
Mean (α) = 1
Mean (α) = 2
0.4
0.3
0.35
0.25
0.3
0.2
P(x)
0.25
P(x)
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0
0
1
2
3
4
5
6
0
1
2
3
4
X
7
8
9
Mean (α) = 10
0.14
0.12
0.1
P(x)
P(x)
6
X
Mean (α) = 5
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
5
0.08
0.06
0.04
0.02
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17
X
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
x
10
Mean (α) = 25
0.09
0.08
0.07
0.06
P(x)
0.05
0.04
0.03
0.02
0.01
0
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38
X
We can see that, as the expected valued increases,
the shape rapidly tends towards the familiar bell of
the normal distribution.
We also note that the most likely value (mode) is
close to the expected value.
Poisson Approximation
The normal distribution can also be used to approximate
the Poisson distribution for large values of .
( - The mean of the Poisson distribution)
If
X ~ Po () then for large values of  > 25
X ~ N (, ) approximately.
Continuity Correction
The Poisson distribution is also a discrete random
variable, whereas the normal distribution is
continuous.
We need to take this into account when we are using
the normal distribution to approximate a Poisson
using a continuity correction.
Example:
A radioactive element disintegrates such that it
follows a Poisson distribution. If the mean number of
particles () emitted is recorded in a 1 second
interval as 69, evaluate the probability of:
(i) Less than 60 particles are emitted in 1 second.
(ii) Between 65 and 75 particles inclusive are emitted
in 1 second.
X ~ Po (69), can be approximated to
X ~ N (69,69) as  > 25.
(i) P (X<60)Po
= P (X<59.5)Norm
= P (z<-1.14)
= 1 – P (z<1.14)
= 1 – 0.8729
= 0.1271
z
59.5  69
69
(ii) P (65X75)Po =
P (64.5<X<75.5)Norm
=
P (-0.54<Z<0.78)
=
0.7823 – (1 – 0.7054)
=
0.7823 – 0.2946
=
0.4877