Editing Geometric distribution (section)

==Definition==

The geometric distribution is the [[discrete probability distribution]] that describes when the first success in an infinite sequence of [[Independent and identically distributed random variables|independent and identically distributed]] [[Bernoulli trial|Bernoulli trials]] occurs. Its [[probability mass function]] depends on its parameterization and [[Support (mathematics)|support]]. When supported on <math>\mathbb{N}</math>, the probability mass function is <math display="block">P(X = k) = (1 - p)^{k-1} p</math> where <math>k = 1, 2, 3, \dotsc</math> is the number of trials and <math>p</math> is the probability of success in each trial.<ref name=":1">{{Cite book |last1=Nagel |first1=Werner |url=https://onlinelibrary.wiley.com/doi/book/10.1002/9781119243496 |title=Probability and Conditional Expectation: Fundamentals for the Empirical Sciences |last2=Steyer |first2=Rolf |date=2017-04-04 |publisher=Wiley |isbn=978-1-119-24352-6 |edition=1st |series=Wiley Series in Probability and Statistics |pages= |language=en |doi=10.1002/9781119243496}}</ref>{{Rp|pages=260–261}}

The support may also be <math>\mathbb{N}_0</math>, defining <math>Y=X-1</math>. This alters the probability mass function into <math display="block">P(Y = k) = (1 - p)^k p</math> where <math>k = 0, 1, 2, \dotsc</math> is the number of failures before the first success.<ref name=":2">{{Cite book |last1=Chattamvelli |first1=Rajan |url=https://link.springer.com/10.1007/978-3-031-02425-2 |title=Discrete Distributions in Engineering and the Applied Sciences |last2=Shanmugam |first2=Ramalingam |publisher=Springer International Publishing |year=2020 |isbn=978-3-031-01297-6 |series=Synthesis Lectures on Mathematics & Statistics |location=Cham |language=en |doi=10.1007/978-3-031-02425-2}}</ref>{{Rp|page=66}}

An alternative parameterization of the distribution gives the probability mass function <math display="block">P(Y = k) = \left(\frac{P}{Q}\right)^k \left(1-\frac{P}{Q}\right)</math> where <math>P = \frac{1-p}{p}</math> and <math>Q = \frac{1}{p}</math>.<ref name=":8" />{{Rp|pages=208–209}}

An example of a geometric distribution arises from rolling a six-sided [[dice|die]] <!-- "die" is the correct singular form of the plural "dice." -->until a "1" appears. Each roll is [[Independence (probability theory)|independent]] with a <math>1/6</math> chance of success. The number of rolls needed follows a geometric distribution with <math>p=1/6</math>.