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Encyclopedia > Geometric distribution
Parameters Support Probability mass function Cumulative distribution function  $0< p leq 1$ success probability (real) $0< p leq 1$ success probability (real) $k in {1,2,3,dots}!$ $k in {0,1,2,3,dots}!$ $(1 - p)^{k-1},p!$ $(1 - p)^{k},p!$ $1-(1 - p)^k!$ $1-(1 - p)^{k+1}!$ $frac{1}{p}!$ $frac{1-p}{p}!$ $leftlceil frac{-log(2)}{log(1-p)} rightrceil!$ (not unique if − log(2) / log(1 − p) is an integer) 1 0 $frac{1-p}{p^2}!$ $frac{1-p}{p^2}!$ $frac{2-p}{sqrt{1-p}}!$ $frac{2-p}{sqrt{1-p}}!$ $6+frac{p^2}{1-p}!$ $6+frac{p^2}{1-p}!$ $-frac{1-p}{p}log_2left(frac{1-p}{p}right)!$ $frac{pe^t}{1-(1-p) e^t}!$ $frac{p}{1-(1-p)e^t}!$ $frac{pe^{it}}{1-(1-p),e^{it}}!$ $frac{p}{1-(1-p),e^{it}}!$

• the probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set { 1, 2, 3, ...}, or
• 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. In the theory of probability and statistics, a Bernoulli trial is an experiment whose outcome is random and can be either of two possible outcomes, called success and failure. ...

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

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 $Pr(X = k) = (1 - p)^{k-1},p,$

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

Equivalently, if the probability of success on each trial is p, then the probability that there are k failures before the first success is $Pr(Y=k) = (1 - p)^k,p,$

for k = 0, 1, 2, 3, ....

In either case, the sequence of probabilities is a geometric sequence. In mathematics, a geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the 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. Two standard six-sided pipped dice with rounded corners. ...

## Contents

The expected value of a geometrically distributed random variable X is 1/p and the variance is (1 − p)/p2: In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ... This article is about mathematics. ... $mathrm{E}(X) = frac{1}{p}, qquadmathrm{var}(X) = frac{1-p}{p^2}.$

Similarly, the expected value of the geometrically distributed random variable Y is (1 − p) / p, and its variance is (1 − p) / p2: $mathrm{E}(Y) = frac{1-p}{p}, qquadmathrm{var}(Y) = frac{1-p}{p^2}.$

Let μ = (1 − p) / p be the expected value of Y. Then the cumulants κn of the probability distribution of Y satisfy the recursion // Cumulants of probability distributions In probability theory and statistics, the cumulants Îºn of the probability distribution of a random variable X are given by In other words, Îºn/n! is the nth coefficient in the power series representation of the logarithm of the moment-generating function. ... $kappa_{n+1} = mu(mu+1) frac{dkappa_n}{dmu}.$

## Parameter estimation

For both variants of the geometric distribution, the parameter p can be estimated by equating the expected value with the sample mean. This is the method of moments, which in this case happens to yield maximum likelihood estimates of p. In mathematics and statistics, the arithmetic mean of a set of numbers is the sum of all the members of the set divided by the number of items in the set. ... In statistics, the method of moments is a method of estimation of population parameters such as mean, variance, median, etc. ... Maximum likelihood estimation (MLE) is a popular statistical method used to make inferences about parameters of the underlying probability distribution from a given data set. ...

Specifically, for the first variant let $k_1,dots,k_n$ be a sample where $k_i geq 1$ for $i=1,dots,n$. Then p can be estimated as A sample is that part of a population which is actually observed. ... $widehat{p} = left(frac1n sum_{i=1}^n k_iright)^{-1}. !$

In Bayesian inference, the Beta distribution is the conjugate prior distribution for the parameter p. If this parameter is given a Beta(αβ) prior, then the posterior distribution is Bayesian inference is statistical inference in which evidence or observations are used to update or to newly infer the probability that a hypothesis may be true. ... Not to be confused with Beta function. ... In Bayesian probability theory, a class of prior probability distributions p(Î¸) is said to be conjugate to a class of likelihood functions p(x|Î¸) if the resulting posterior distributions p(Î¸|x) are in the same family as p(Î¸). For example, the Gaussian family is conjugate to itself (or self-conjugate... A prior probability is a marginal probability, interpreted as a description of what is known about a variable in the absence of some evidence. ... In Bayesian probability theory, the posterior probability is the conditional probability of some event or proposition, taking empirical data into account. ... $p sim mathrm{Beta}left(alpha+n, beta+sum_{i=1}^n (k_i-1)right). !$

The posterior mean E[p] approaches the maximum likelihood estimate $widehat{p}$ as α and β approach zero.

In the alternative case, let $k_1,dots,k_n$ be a sample where $k_i geq 0$ for $i=1,dots,n$. Then p can be estimated as $widehat{p} = left(1 + frac1n sum_{i=1}^n k_iright)^{-1}. !$

The posterior distribution of p given a Beta(αβ) prior is $p sim mathrm{Beta}left(alpha+n, beta+sum_{i=1}^n k_iright). !$

Again the posterior mean E[p] approaches the maximum likelihood estimate $widehat{p}$ as α and β approach zero.

## Other properties $G_X(s) = frac{s,p}{1-s,(1-p)}, !$ $G_Y(s) = frac{p}{1-s,(1-p)}, quad |s| < (1-p)^{-1}. !$
• Like its continuous analogue (the exponential distribution), the geometric distribution is memoryless. That means that if you intend to repeat an experiment until the first success, then, given that the first success has not yet occurred, the conditional probability distribution of the number of additional trials does not depend on how many failures have been observed. The die one throws or the coin one tosses does not have a "memory" of these failures. The geometric distribution is in fact the only memoryless discrete distribution.
• Among all discrete probability distributions supported on {1, 2, 3, ... } with given expected value μ, the geometric distribution X with parameter p = 1/μ is the one with the largest entropy.
• The geometric distribution of the number Y of failures before the first success is infinitely divisible, i.e., for any positive integer n, there exist independent identically distributed random variables Y1, ..., Yn whose sum has the same distribution that Y has. These will not be geometrically distributed unless n = 1; they follow a negative binomial distribution.
• The decimal digits of the geometrically distributed random variable Y are a sequence of independent (and not identically distributed) random variables. For example, the hundreds digit D has this probability distribution: $Pr(D=d) = {q^{100d} over {1 + q^{100} + q^{200} + cdots + q^{900}}},$
where q = 1 − p, and similarly for the other digits, and, more generally, similarly for numeral systems with other bases than 10. When the base is 2, this shows that a geometrically distributed random variable can be written as a sum of independent random variables whose probability distributions are indecomposable.

PGF redirects here, for other uses see PGF (disambiguation). ... In probability theory and statistics, the exponential distributions are a class of continuous probability distribution. ... In probability theory, memorylessness is a property of certain probability distributions: the exponential distributions and the geometric distributions. ... In statistics and information theory, a maximum entropy probability distribution is a probability distribution whose entropy is larger than (or equal to) that of all other members of a specified class of distributions. ... The concept of infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). ... In probability and statistics the negative binomial distribution is a discrete probability distribution. ... This article is about different methods of expressing numbers with symbols. ... In probability theory, an indecomposable distribution is any probability distribution that cannot be represented as the distribution of the sum of two or more non-constant independent random variables. ...

## Related distributions $Z = sum_{m=1}^r Y_m$
follows a negative binomial distribution with parameters r and p.
• If Y1,...,Yr are independent geometrically distributed variables (with possibly different success parameters p(m)), then their minimum $W = min_{m} Y_m,$
is also geometrically distributed, with parameter p given by
 1 − ∏ (1 − p(m)). m
• Suppose 0 < r < 1, and for k = 1, 2, 3, ... the random variable Xk has a Poisson distribution with expected value rk/k. Then $sum_{k=1}^infty k,X_k$
has a geometric distribution taking values in the set {0, 1, 2, ...}, with expected value r/(1 − r).
• The exponential distribution is the continuous analogue of the geometric distribution. If a random variable with an exponential distribution is rounded up to the next integer then the result is a discrete random variable with a geometric distribution.

In probability and statistics the negative binomial distribution is a discrete probability distribution. ... The largest and the smallest element of a set are called extreme values, or extreme records. ... In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate, and are independent of the time since the last event. ... In probability theory and statistics, the exponential distributions are a class of continuous probability distribution. ... The floor function The ceiling function In mathematics, the floor and the ceiling functions are two functions which convert arbitrary real numbers to close integers. ...

• Coupon collector's problem Results from FactBites:

 Geometric distribution - Wikipedia, the free encyclopedia (471 words) In either case, the sequence of probabilities is a geometric sequence. } and is a geometric distribution with p = 1/6. The geometric distribution Y is a special case of the negative binomial distribution, with r = 1.
 Probability distribution - Wikipedia, the free encyclopedia (1449 words) A probability distribution is a special case of the more general notion of a probability measure, which is a function that assigns probabilities satisfying the Kolmogorov axioms to the measurable sets of a measurable space. The rectangular distribution is a uniform distribution on [-1/2,1/2]. The triangular distribution on [a, b], a special case of which is the distribution of the sum of two uniformly distributed random variables (the convolution of two uniform distributions).
More results at FactBites »

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