Difference between revisions of "Geometric distribution"
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| − | The distribution of a discrete random variable assuming non-negative integral values | + | The distribution of a discrete random variable assuming non-negative integral values $ m = 0, 1 \dots $ |
| + | 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 | + | the mathematical expectation is $ q/p $; |
| + | the variance is $ q/ p ^ {2} $; | ||
| + | the generating function is | ||
| − | + | $$ | |
| + | P ( t) = | ||
| + | \frac{p}{1 - qt } | ||
| + | . | ||
| + | $$ | ||
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/g044230a.gif" /> | <img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/g044230a.gif" /> | ||
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Figure: g044230a | Figure: g044230a | ||
| − | A geometric distribution of probability | + | A geometric distribution of probability $ p _ {m} $. |
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/g044230b.gif" /> | <img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/g044230b.gif" /> | ||
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Figure: g044230b | Figure: g044230b | ||
| − | The distribution function | + | 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 | + | 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. | ||
Latest revision as of 19:41, 5 June 2020
2020 Mathematics Subject Classification: Primary: 60E99 [MSN][ZBL]
The distribution of a discrete random variable assuming non-negative integral values $ m = 0, 1 \dots $ 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
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