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(MSC|60E99 Category:Distribution theory)
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[[Category:Distribution theory]]
 
[[Category:Distribution theory]]
  
The distribution of a discrete random variable assuming non-negative integral values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044230/g0442301.png" /> with probabilities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044230/g0442302.png" />, where the distribution parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044230/g0442303.png" /> is a number in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044230/g0442304.png" />. The characteristic function is
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The distribution of a discrete random variable assuming non-negative integral values $  m = 0, 1 \dots $
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with probabilities $  p _ {m} = pq  ^ {m} $,  
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where the distribution parameter $  p = 1 - q $
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is a number in $  ( 0, 1) $.  
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The characteristic function is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044230/g0442305.png" /></td> </tr></table>
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$$
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f ( t)  =
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\frac{p}{1 - qe  ^ {it} }
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,
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$$
  
the mathematical expectation is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044230/g0442306.png" />; the variance is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044230/g0442307.png" />; the generating function is
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the mathematical expectation is $  q/p $;  
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the variance is $  q/ p  ^ {2} $;  
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the generating function is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044230/g0442308.png" /></td> </tr></table>
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$$
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P ( t)  =
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\frac{p}{1 - qt }
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.
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$$
  
 
<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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044230/g0442309.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044230/g04423010.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044230/g04423011.png" /> and a probability of failure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044230/g04423012.png" /> has a geometric distribution. The name originates from the geometric progression which generates such a distribution.
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The random variable equal to the number of independent trials prior to the first successful outcome with a probability of success $  p $
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and a probability of failure $  q $
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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
This article was adapted from an original article by V.M. Kalinin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article