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May 11’09
THE POISSON PROBABILITY DISTRIBUTION
#7
This
discrete
distribution, named for French mathematician S.D.Poisson (17811840),
is used extensively as a probability model in biology and medicine.
If
x
is a number of occurrences of some random event in an interval of time or space
(or some volume of matter), the Poisson probability that
x
will occur is given by
f(x) = e
–
λ
λ
x
/x!
(
x = 0, 1, 2, …
)
(7.1)
The Greek letter
λ
(lambda) is called the
parameter of the distribution
and is the average
number of occurrences of the random event in an interval (or volume). The symbol
e
in
(7.1) is the constant
2.71828
…
It can be shown that
f(x) ≥ 0
for every
x
and
Σ
x
f(x) = 1
so the distribution (7.1) satisfies the requirements for a probability distribution.
Some statistical problems require the evaluating of a
cumulative
Poisson probability
function
f(X ≤ x)
which is defined as follows:
f(X ≤ x)
=
Σ
f(X)
(7.1a)
all
X ≤ x
The function (7.1a) represents the probability that a number
X
of occurrences of the
random event of interest in an interval of time or space (or some volume of matter) does
not exceed the given number
x
. In the expanded form,
x
x
f(X ≤ x)
=
Σ
f(i) =
Σ
e
–
λ
λ
i
/ i!
(7.1b)
i=0
i=0
A sum of probabilities
P(X ≥ x) + P(X ≤
x
– 1
) = 1
for both binomial and Poisson
discrete
probability distributions because the events
X ≥ x
and
X ≤
x
–1
are
complementary
to each other; also
P(k ≤ X ≤
m) =
P(X ≤
m) –
P(X ≤
k
– 1
).
An interesting feature of the Poisson distribution is the fact that both
the mean
µ
and the variance
σ
2
are equal
to the
parameter
λ
of the distribution
.
The Poisson process
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 Spring '05
 Kzaer
 Poisson Distribution, Probability

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