The Poisson distribution describes the probability of a number of independent events occurring in a fixed interval of time or space when the average rate of occurrences is known. It can be used when there are a large number of trials with a small probability of success, like the number of deaths from horse kicks in the Army each year. The mean number of successes is equal to the number of trials multiplied by the probability of success. The Poisson distribution becomes applicable when the number of trials approaches infinity and the probability of success approaches zero while the mean remains constant. It is used to model random events that are expected to occur with a consistent average rate.

1. The Poisson
Distribution

2. The Poisson Distribution
Overview
• When there is a large number of
trials, but a small probability of
success, binomial calculation
becomes impractical
– Example: Number of deaths
from horse kicks in the Army in
different years
• The mean number of successes
from n trials is µ = np
– Example: 64 deaths in 20 years
from thousands of soldiers
Simeon D. Poisson (1781-1840)

3. The Poisson Distribution
Overview
• If we substitute µ/n for p, and let n tend to infinity, the
binomial distribution becomes the Poisson distribution:
P(x) =
e -µµx
x!

4. The Poisson Distribution
Overview
• Poisson distribution is applied where random events in space
or time are expected to occur
• Deviation from Poisson distribution may indicate some degree
of non-randomness in the events under study
• Investigation of cause may be of interest

5. The Poisson Distribution
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