Geometric distribution

2010 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.