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Geometric distribution

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2020 Mathematics Subject Classification: Primary: 60E99 [MSN][ZBL]

The distribution of a discrete random variable assuming non-negative integral values with probabilities p _ {m} = pq ^ {m} , where the distribution parameter p = 1 - q is a number in ( 0, 1) . The characteristic function is

f ( t) = \frac{p}{1 - qe ^ {it} } ,

the mathematical expectation is q/p ; the variance is q/ p ^ {2} ; the generating function is

P ( t) = \frac{p}{1 - qt } .

Figure: g044230a

A geometric distribution of probability p _ {m} .

Figure: g044230b

The distribution function ( p = 0.2) .

The random variable equal to the number of independent trials prior to the first successful outcome with a probability of success p and a probability of failure q has a geometric distribution. The name originates from the geometric progression which generates such a distribution.

How to Cite This Entry:
Geometric distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Geometric_distribution&oldid=47089
This article was adapted from an original article by V.M. Kalinin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article