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{{MSC|60E99}}

[[Category:Distribution theory]]

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

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

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/g044230a.gif" />

Figure: g044230a

A geometric distribution of probability  $  p _ {m} $.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/g044230b.gif" />

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.