Geometric Distribution

A Geometric Distribution is a discrete probability distribution, it is a discrete analog of the exponential distribution.

• Context
  • For k=0, 1, 2, ..., its probability density function is given by

[math]\displaystyle{ P(k)=p(1-p)^k=pq^k, }[/math]

and the distribution function is

[math]\displaystyle{ F(k)=\sum_{k=0}^{n}P(k)=1-q^{n+1} }[/math]

• Example(s):
  • a Compound Poisson Distribution.
  • …
• Counter-Example(s):
  • a Poisson Distribution.
• See: Hypergeometric Probability Distribution, Discrete Probability Distribution.

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References

2016

• (Wikipedia, 2014) ⇒ https://www.wikiwand.com/en/Geometric_distribution Retrieved 2016-06-11
  • In probability theory and statistics, the geometric distribution is either of two discrete probability distributions:
    • The probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set { 1, 2, 3, ...}
    • The probability distribution of the number Y = X − 1 of failures before the first success, supported on the set { 0, 1, 2, 3, ... }

Which of these one calls "the" geometric distribution is a matter of convention and convenience.

These two different geometric distributions should not be confused with each other. Often, the name shifted geometric distribution is adopted for the former one (distribution of the number X); however, to avoid ambiguity, it is considered wise to indicate which is intended, by mentioning the support explicitly.

It’s the probability that the first occurrence of success requires k number of independent trials, each with success probability p. If the probability of success on each trial is p, then the probability that the kth trial (out of k trials) is the first success is

[math]\displaystyle{ \Pr(X = k) = (1-p)^{k-1}\,p\, }[/math]

for k = 1, 2, 3, ....

The above form of geometric distribution is used for modeling the number of trials up to and including the first success. By contrast, the following form of the geometric distribution is used for modeling the number of failures until the first success:

[math]\displaystyle{ \Pr(Y=k) = (1 - p)^k\,p\, }[/math]

for k = 0, 1, 2, 3, ....

In either case, the sequence of probabilities is a geometric sequence.

For example, suppose an ordinary die is thrown repeatedly until the first time a "1" appears. The probability distribution of the number of times it is thrown is supported on the infinite set { 1, 2, 3, ... } and is a geometric distribution with p = 1/6.

2008

• (Upton & Cook, 2008)

2006

• (Dubnicka, 2006f) ⇒ Suzanne R. Dubnicka. (2006). “Special Discrete Distributions - Handout 6." Kansas State University, Introduction to Probability and Statistics I, STAT 510 - Fall 2006.
  • TERMINOLOGY : Imagine an experiment where Bernoulli trials are observed. If X denotes the trial on which the first success occurs, then X is said to follow a geometric distribution with parameter p, the probability of success on any one trial, 0 < p < 1. This is sometimes written as X Geo(p). The pmf for X is given by pX(x) =
    • (1 − p)x−1p, x = 1, 2, 3, . . .
    • 0, otherwise.

1999

• (Weisstein, 1999) ⇒ Eric W. Weisstein. (1999) "Geometric Distribution." From MathWorld -- A Wolfram Web Resource. http://mathworld.wolfram.com/GeometricDistribution.html
  • The geometric distribution is a discrete distribution for n=0, 1, 2, … having probability density function

[math]\displaystyle{ P(n)=p(1-p)^n=pq^n, }[/math]

where [math]\displaystyle{ 0\lt p\lt 1, q=1-p }[/math] and distribution function is

[math]\displaystyle{ D(n)=\sum_{k=0}^{n}P(k)=1-q^{n+1} }[/math]

The geometric distribution is the only discrete memoryless random distribution. It is a discrete analog of the exponential distribution.

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