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Poisson Distribution
The Poisson distribution with parameter l is the discrete probability distribution
of a random variable X with the following probability mass
function:
The Poisson distribution is often used to model the
probability distribution of the number of outcomes that are
labeled 'successes' during a given time interval or within a
specified region. The time interval involved could have a variety
of lengths, e.g., a second, minute, hour, day, year, and
multiples thereof. The region in question could be a line
segment, an area, a volume, or some n-dimensional space, where n
is an integer. Experiments of this type are known as
Poisson experiments, and they possess the following
characteristics:
- The number of outcomes occurring in any given time
interval or region is independent of the number of
outcomes occurring in any other disjoint time interval or
region.
- The probability of a single outcome occurring in a very
short time interval or very small region is proportional
to the length of the time interval or the size of the
region. This value is not affected by the number of
outcomes occurring outside this particular time interval
or region.
- The probability of having more than one outcome occurring
in a very short time interval or very small region is
negligible.
The mean and variance of a random variable X that has a
Poisson distribution with parameter l
are both equal to each other and to the parameter l, i.e.,
E(X) = Var(X) = l
EX. In one particular autobiography of a professional athlete,
there are an average of 15 spelling errors per page. If the
Poisson distribution is used to model the probability
distribution of the number of errors per page, then the random
variable X, the number of errors per page, has a Poisson
distribution with l = 15. The
probability that there are no errors on a page is e-15150
= 3.06 x 10-7.
(How much would you pay for a book with such a lack of
spelling ability?)
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