Geometric distribution
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.
Geometric distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Geometric_distribution&oldid=47089