Page 42 - Computational Statistics Handbook with MATLAB

28 Computational Statistics Handbook with MATLAB

second condition specifies that the number of events in one interval does not
affect the number of events in other intervals. The first condition states that
the counting starts at time t = 0 . The expected value of Nt() is given by

[
EN t()] = λt .

Example 2.2
In preparing this text, we executed the spell check command, and the editor
reviewed the manuscript for typographical errors. In spite of this, some mis-
takes might be present. Assume that the number of typographical errors per
page follows the Poisson distribution with parameter λ = 0.25 . We calculate
the probability that a page will have at least two errors as follows:

(
PX ≥ 2) = 1 – { PX = 0) + PX = 1)} = 1 – e – 0.25 – e – 0.25 0.25 ≈ 0.0265 .
(
(
We can get this probability using the MATLAB Statistics Toolbox function
(
poisscdf. Note that PX =( 0) + PX = 1) is the Poisson cumulative distri-
bution function for a = 1 (see Equation 2.4), which is why we use 1 as the
argument to poisscdf.
prob = 1-poisscdf(1,0.25);

Example 2.3
Suppose that accidents at a certain intersection occur in a manner that satis-
fies the conditions for a Poisson process with a rate of 2 per week ( λ = 2 ).
What is the probability that at most 3 accidents will occur during the next 2
weeks? Using Equation 2.26, we have

3
PN 2() ≤ 3) = ∑ PN 2() = k) .
(
(
k = 0
Expanding this out yields

2 3
(
PN 2() ≤ 3) = e + 4e + 4 – 4 4 -----e ≈ 0.4335 .
–
4
–
4
4
–
-----e +
2! 3!
As before, we can use the poisscdf function with parameter given by
⋅
λt = 22 .
prob = poisscdf(3,2*2);

© 2002 by Chapman & Hall/CRC