Geometric Distribution

In probability theory and statistics, the geometric distribution is either of two discrete probability distributions:

  • The probability distribution of the number of X 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.

Geometric
Probability mass function
Cumulative distribution function
Parameters success probability (real) success probability (real)
Support
Probability mass function (pmf)
Cumulative distribution function (cdf)
Mean
Median (not unique if is an integer) (not unique if is an integer)
Mode
Variance
Skewness
Excess kurtosis
Entropy
Moment-generating function (mgf) ,
for
Characteristic function

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 range explicitly.

It’s the probability that the first occurrence of success require 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

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

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

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.

Read more about Geometric DistributionMoments and Cumulants, Parameter Estimation, Other Properties, Related Distributions

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