View source for Geometric distribution

[[category:Discrete distributions]]
[[category:Semi-bounded distributions]]
[[category:Unimodal distributions]]
[[category:Univariate distributions]]
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The geometric distribution describes the number of independent [[Bernoulli]] trials until the first successful outcome occurs -- for example, the number of coin tosses until the first heads. The parameter «p» is the probability of success on any given trial.

<center>[[image:Geometric(0.4).png]]</center>

== Functions ==
=== Geometric(p) ===
The [[:category:Distribution Functions|distribution function]]. Use this to define an uncertain variable that represents a number of events that occur until a success occurs, where «p» is the probability of success on each trial.

=== <div id="ProbGeometric">Prob{{Release||5.1|_}}Geometric(k, p)</div> ===
{{Release||5.1|To use, add the [[Distribution Densities Library]] to your model.}}

Returns the probability that «k» or fewer events occur until a success is seen. This is the [[Category:Discrete probability functions|analytic discrete probability function]]. It returns 

:<math>p(k) = p (1-p)^{k-1}</math>

=== <div id="CumGeometric">CumGeometric(k ,p)</div> ===
{{Release||5.1|To use, add the [[Distribution Densities Library]] to your model.}}

The [[:category:Cumulative distribution functions|analytic cumulative probability function]] for a [[Geometric distribution]]. This is the probability that the actual number of events that occur until a success occurs is less than or equal to «k», given that at each trial there is a «p» probability of success. The value is equal to

:<math>F(k) = 1 - (1-p)^k</math>

=== <div id="CumGeometricInv">CumGeometricInv(q, p)</div> ===
{{Release||5.1|To use, add the [[Distribution Densities Library]] to your model.}}

The [[:category:Inverse cumulative probability functions|inverse of the cumulative probability function]] for the [[Geometric]](p) distribution, also know as the ''quantile function''.

Computes the smallest number of independent [[Bernoulli]] trials, ''k'', such that the probability of seeing at most ''k'' of failures before the first success is greater than or equal to «u», where the probability of success of each trial is «p».

== Statistics ==
The [[Geometric distribution]] has the following statistics
* [[Min]] = 1
* [[Mean]] = 1/p
* [[Mode]] = 1
* [[Variance]] = <math> (1-p)/p^2</math>
* [[Skewness]] = <math>{{2-p}\over{\sqrt{1-p}}}</math>

== Examples ==
An aspiring gymnast catches her jaeger (a release move on uneven bars) at practice 40% of the time.  Her coach wants her to successfully catch at least one during practice 95% of the time (i.e., in 95% of her practices, she should catch at least one).  How many repetitions should the coach insist on during each practice?

:<code>CumGeometricInv(95%, 40%) &rarr; 6</code>

The actual success rate if she makes 6 attempts every practice should be

:<code>CumGeometric(6, 40%) &rarr; 95.33%</code>

If you enter a lottery every day of the year, where each entry has odds of 1 chance in 1M of winning, what is the probability that you will win within one year?  How about within 10 years?
:<code>CumGeometric(365, 1/1M) &rarr; 3.649e-004</code>
:<code>CumGeometric(3653, 1/1M) &rarr; 3.646e-003</code>

What is the probability of rolling doubles ten times in a row with a pair of fair dice?  To encode this, treat a success as a non-double roll, which has a probability of 5/6, so the answer is given by
:<code>1 - CumGeometric(10, 5/6) &rarr; 1.654e-008</code>

John is a baseball player, who hits the ball on 10% of his swings.  What is the probability he gets a hit during his next three swings?

:<code>CumGeometric(3, 10%)</code>

== History ==
The analytic functions (ProbGeometric, CumGeometric, and CumGeometricInv) were included as built-in functions in [[Analytica 5.2]].

The analytic functions were provided in the [[Distribution Densities Library]] for the first time in the 4.4.3 patch release. 

== See Also ==
* [[Bernoulli distribution]]
* [[Binomial distribution]]
* [[HyperGeometric distribution]]