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Binomial distribution
The binomial distribution applies when there are two
possible outcomes. You know the probability of obtaining
either outcome (traditionally called "success" and "failure")
and want to know the chance of obtaining a certain number
of successes in a certain number of trials.
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Poisson distribution
The Poisson distribution applies when you are counting the
number of objects in a certain volume or the number of
events in a certain time period. You know the average
number of counts, and wish to know the chance of actually
observing various numbers of objects or events.
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Gaussian distribution
The Gaussian distribution applies when the outcome is expressed
as a number that can have a fractional value. If there are
numerous reasons why any particular measurement is different
than the mean, the distribution of measurements will tend to
follow a Gaussian bell-shaped distribution. If you know the
mean and SD of this distribution, you can compute the fraction
of the population that is greater (or less) than any particular
value.
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