In probability theory and statistics, the geometric distribution is either of two discrete probability distributions: Image File history File links Geometricpdf. ...
Image File history File links Geometriccdf. ...
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
In mathematics, the support of a realvalued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ...
In probability theory, a probability mass function (abbreviated pmf) gives the probability that a discrete random variable is exactly equal to some value. ...
In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a realvalued random variable, X. For every real number x, the cdf is given by where the righthand side represents the probability that the random variable X takes on a value less than...
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...
This article is about the statistical concept. ...
In statistics, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. ...
This article is about mathematics. ...
Example of experimental data with nonzero skewness (gravitropic response of wheat coleoptiles, 1,790) In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a realvalued random variable. ...
The far red light has no effect on the average speed of the gravitropic reaction in wheat coleoptiles, but it changes kurtosis from platykurtic to leptokurtic (0. ...
Claude Shannon In information theory, the Shannon entropy or information entropy is a measure of the uncertainty associated with a random variable. ...
In probability theory and statistics, the momentgenerating function of a random variable X is wherever this expectation exists. ...
In probability theory, the characteristic function of any random variable completely defines its probability distribution. ...
Probability theory is the branch of mathematics concerned with analysis of random phenomena. ...
This article is about the field of statistics. ...
In mathematics, a probability distribution is called discrete, if it is fully characterized by a probability mass function. ...
 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 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 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 sixsided pipped dice with rounded corners. ...
Moments and cumulants
The expected value of a geometrically distributed random variable X is 1/p and the variance is (1 − p)/p^{2}: 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. ...
Similarly, the expected value of the geometrically distributed random variable Y is (1 − p) / p, and its variance is (1 − p) / p^{2}: Let μ = (1 − p) / p be the expected value of Y. Then the cumulants κ_{n} 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 momentgenerating function. ...
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 be a sample where for . Then p can be estimated as A sample is that part of a population which is actually observed. ...
In Bayesian inference, the Beta distribution is the conjugate prior distribution for the parameter p. If this parameter is given a Beta(α, β) 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(xÎ¸) 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 selfconjugate...
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. ...
The posterior mean E[p] approaches the maximum likelihood estimate as α and β approach zero. In the alternative case, let be a sample where for . Then p can be estimated as The posterior distribution of p given a Beta(α, β) prior is Again the posterior mean E[p] approaches the maximum likelihood estimate as α and β approach zero.
Other properties 
 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 Y_{1}, ..., Y_{n} 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:

 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 nonconstant independent random variables. ...
Related distributions  The geometric distribution Y is a special case of the negative binomial distribution, with r = 1. More generally, if Y_{1},...,Y_{r} are independent geometrically distributed variables with parameter p, then

 follows a negative binomial distribution with parameters r and p.
 If Y_{1},...,Y_{r} are independent geometrically distributed variables (with possibly different success parameters p^{(m)}), then their minimum

 is also geometrically distributed, with parameter p given by
 Suppose 0 < r < 1, and for k = 1, 2, 3, ... the random variable X_{k} has a Poisson distribution with expected value r^{k}/k. Then

 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. ...
See also  Coupon collector's problem
External links Probability distributions   Discrete univariate with finite support      Discrete univariate with infinite support      Continuous univariate supported on a bounded interval      Continuous univariate supported on a semiinfinite interval      Continuous univariate supported on the whole real line            PlanetMath is a free, collaborative, online mathematics encyclopedia. ...
MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...
A probability distribution describes the values and probabilities that a random event can take place. ...
A probability distribution describes the values and probabilities that a random event can take place. ...
A logarithmic scale bar. ...
In probability theory and statistics, the Bernoulli distribution, named after Swiss scientist Jakob Bernoulli, is a discrete probability distribution, which takes value 1 with success probability and value 0 with failure probability . ...
In probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. ...
Often confused with the multinomial distribution. ...
// In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the number of successes in a sequence of n draws from a finite population without replacement. ...
In probability theory and statistics, the Rademacher distribution is a discrete probability distribution. ...
In probability theory and statistics, the discrete uniform distribution is a discrete probability distribution that can be characterized by saying that all values of a finite set of possible values are equally probable. ...
Zipfs law, an empirical law formulated using mathematical statistics, refers to the fact that many types of data studied in the physical and social sciences can be approximated with a Zipfian distribution, one of a family of related discrete power law probability distributions. ...
In probability theory and statistics, the ZipfMandelbrot law is a discrete probability distribution. ...
A probability distribution describes the values and probabilities that a random event can take place. ...
In physics, the Boltzmann distribution predicts the distribution function for the fractional number of particles Ni / N occupying a set of states i which each has energy Ei: where is the Boltzmann constant, T is temperature (assumed to be a sharply welldefined quantity), is the degeneracy, or number of...
In probability theory, a compound Poisson distribution is the probability distribution of a Poissondistibuted number of independent identicallydistributed random variables. ...
The discrete phasetype distribution is a probability distribution that results from a system of one or more interrelated geometric distributions occurring in sequence, or phases. ...
In mathematics, the GaussKuzmin distribution gives the probability distribution of the occurrence of a given integer in the continued fraction expansion of an arbitrary real number. ...
In probability and statistics, the logarithmic distribution (also known as the logarithmic series distribution) is a discrete probability distribution. ...
In probability and statistics the negative binomial distribution is a discrete probability distribution. ...
In the parabolic fractal distribution, the logarithm of the frequency or size of entities in a population is a quadratic polynomial of the logarithm of the rank. ...
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. ...
The Skellam distribution is the discrete probability distribution of the difference N1 âˆ’ N2 of two correlated or uncorrelated random variables N1 and N2 having Poisson distributions with different expected values Î¼1 and Î¼2. ...
In probability and statistics, the YuleSimon distribution is a discrete probability distribution. ...
In probability theory and statistics, the zeta distribution is a discrete probability distribution. ...
In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ...
Not to be confused with Beta function. ...
In probability theory and statistics, Kumaraswamys double bounded distribution is as versatile as the Beta distribution, but much simpler to use especially in simulation studies as it has a simple closed form solution for both its pdf and cdf. ...
In probability theory and statistics, the raised cosine distribution is a probability distribution supported on the interval []. The probability density function is for and zero otherwise. ...
In probability theory and statistics, the triangular distribution is a continuous probability distribution with lower limit a, mode c and upper limit b. ...
In probability theory and statistics, the Uquadratic distribution is a continuous probability distribution defined by a unique quadratic function with lower limit a and upper limit b. ...
In probability theory and statistics, the continuous uniform distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distributions support are equally probable. ...
The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution supported on the interval [âˆ’R, R] the graph of whose probability density function f is a semicircle of radius R centered at (0, 0) and then suitably normalized (so that it is really a semiellipse...
In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ...
A Beta Prime Distribution is a distribution with probability function: where is a Beta function. ...
This article is about the mathematics of the chisquare distribution. ...
A phasetype distribution is a probability distribution that results from a system of one or more interrelated Poisson processes occurring in sequence, or phases. ...
The Erlang distribution is a continuous probability distribution with wide applicability primarily due to its relation to the exponential and Gamma distributions. ...
In probability theory and statistics, the exponential distributions are a class of continuous probability distribution. ...
In statistics and probability, the Fdistribution is a continuous probability distribution. ...
This article does not cite its references or sources. ...
The folded normal distribution is a probability distribution related to the normal distribution. ...
In probability theory and statistics, the gamma distribution is a twoparameter family of continuous probability distributions. ...
In probability theory and statistics, the generalized extreme value distribution (GEV) is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, FrÃ©chet and Weibull families also known as type I, II and III extreme value distributions. ...
In probability theory, the Generalized inverse Gaussian distribution (GIG) is a probability distribution with probability density function It is used extensively in geostatistics, statistical linguistics, finance, etc. ...
In probability theory and statistics, the halflogistic distribution is a continuous probability distributionâ€”the distribution of the absolute value of a random variable following the logistic distribution. ...
In statistics, Hotellings Tsquare statistic, named for Harold Hotelling, is a generalization of Students t statistic that is used in multivariate hypothesis testing. ...
In probability theory, a hyperexponential distribution is a continuous distribution such that the probability density function of the random variable X is given by: Where is an exponentially distributed random variable with rate parameter , and is the probability that X will take on the form of the exponential distribution...
The hypoexponential distribution is a generalization of Erlang distribution in the sense that the n exponential distributions may have different rates. ...
In probability and statistics, the inversechisquare distribution is the probability distribution of a random variable whose inverse has a chisquare distribution. ...
The scaleinversechisquare distribution arises in Bayesian statistics (spam filtering in particular). ...
The probability density function of the inverse Gaussian distribution is given by The Wald distribution is simply another name for the inverse Gaussian distribution. ...
The inverse gamma distribution has the probability density function over the support with shape parameter and scale parameter . ...
In probability theory and statistics, the LÃ©vy distribution, named after Paul Pierre LÃ©vy, is one of the few distributions that are stable and that have probability density functions that are analytically expressible. ...
In probability and statistics, the lognormal distribution is the singletailed probability distribution of any random variable whose logarithm is normally distributed. ...
The Maxwellâ€“Boltzmann distribution is a probability distribution with applications in physics and chemistry. ...
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There are very few or no other articles that link to this one. ...
In probability theory and statistics, the noncentral chisquare or noncentral distribution is a generalization of the chisquare distribution. ...
The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution found in a large number of realworld situations. ...
A phasetype distribution is a probability distribution that results from a system of one or more interrelated Poisson processes occurring in sequence, or phases. ...
In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution. ...
The relativistic Breitâ€“Wigner distribution (after Gregory Breit and Eugene Wigner) is a continuous probability distribution with the following probability density function [1]: It is most often used to model resonances (i. ...
In probability theory and statistics, the Rice distribution distribution is a continuous probability distribution. ...
The shifted Gompertz distribution is the distribution of the largest order statistic of two independent random variables which are distributed exponential and Gompertz with parameters b and b and respectively. ...
In probability and statistics, the truncated normal distribution is the probability distribution of a normally distributed random variable whose value is either bounded below or above (or both). ...
In probability theory, the Type2 Gumbel distribution function is for . Based on gslref_19. ...
In probability theory and statistics, the Weibull distribution (named after Waloddi Weibull) is a continuous probability distribution with the probability density function where and is the shape parameter and is the scale parameter of the distribution. ...
This article or section is in need of attention from an expert on the subject. ...
In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ...
The CauchyLorentz distribution, named after Augustin Cauchy, is a continuous probability distribution with probability density function where x0 is the location parameter, specifying the location of the peak of the distribution, and Î³ is the scale parameter which specifies the halfwidth at halfmaximum (HWHM). ...
This article needs cleanup. ...
The exponential power distribution, also known as the generalized error distribution, takes a scale parameter a and exponent b. ...
Fishers zdistribution is the distribution of half the logarithm of a F distribution variate: It was first described by Ronald Fisher in a paper delivered at the International Mathematical Congress of 1924 in Toronto, entitled On a distribution yielding the error functions of several wellknown statistics. Nowadays...
In probability theory and statistics the Gumbel distribution (named after Emil Julius Gumbel (1891â€“1966)) is used to find the minimum (or the maximum) of a number of samples of various distributions. ...
The generalised hyperbolic distribution is a continuous probability distribution defined by the probability density function where is the modified Bessel function of the second kind. ...
In probability theory and statistics, the hyperbolic secant distribution is a continuous probability distribution whose probability density function and characteristic function are proportional to the hyperbolic secant function. ...
The probability distribution for Landau random variates is defined analytically by the complex integral, For numerical purposes it is more convenient to use the following equivalent form of the integral, From GSL manual, used under GFDL. ...
In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after PierreSimon Laplace. ...
In probability theory, a LÃ©vy skew alphastable distribution or just stable distribution, developed by Paul LÃ©vy, is a probability distribution where sums of independent identically distributed random variables have the same distribution as the original. ...
In probability theory and statistics, the logistic distribution is a continuous probability distribution. ...
The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ...
In probability theory and statistics, the normalgamma distribution is a fourparameter family of continuous probability distributions. ...
The normalinverse Gaussian distribution is continuous probability distribution that is defined as the normal variancemean mixture where the mixing density is the inverse Gaussian distribution. ...
In probability and statistics, the tdistribution or Students tdistribution is a probability distribution that arises in the problem of estimating the mean of a normally distributed population when the sample size is small. ...
In probability theory, the Type1 Gumbel distribution function is for . Reference Taken from the gslref_19. ...
The variancegamma distribution is continuous probability distribution that is defined as the normal variancemean mixture where the mixing density is the gamma distribution. ...
In spectroscopy, the Voigt profile is a spectral line profile named after Woldemar Voigt and found in all branches of spectroscopy in which a spectral line is broadened by two types of mechanisms, one of which alone would produce a Doppler profile, and the other of which would produce a...
A probability distribution describes the values and probabilities that a random event can take place. ...
In population genetics, Ewenss sampling formula, introduced by Warren Ewens, states that under certain conditions (specified below), if a random sample of n gametes is taken from a population and classified according to the gene at a particular locus then the probability that there are a1 alleles represented once...
In probability theory, the multinomial distribution is a generalization of the binomial distribution. ...
The multivariate Polya distribution, also called the Dirichlet compound multinomial distribution, is a compound probability distribution, where a probability vector p is drawn from a Dirichlet distribution and a set of discrete samples x is drawn from the multinomial distribution with probability vector p. ...
Several images of the probability density of the Dirichlet distribution when K=3 for various parameter vectors Î±. Clockwise from top left: Î±=(6, 2, 2), (3, 7, 5), (6, 2, 6), (2, 3, 4). ...
In statistics, the generalized Dirichlet distribution (GD) is a generalization of the Dirichlet distribution with a more general covariance structure and twice the number of parameters. ...
In probability theory and statistics, a multivariate normal distribution, also sometimes called a multivariate Gaussian distribution, is a specific probability distribution, which can be thought of as a generalization to higher dimensions of the onedimensional normal distribution (also called a Gaussian distribution). ...
In statistics, a multivariate Student distribution is a multivariate generalization of the Students tdistribution. ...
A probability distribution describes the values and probabilities that a random event can take place. ...
In statistics, the Inverse Wishart distribution, also the inverse Wishart distribution and inverted Wishart distribution is a probability density function defined on matrices. ...
The matrix normal distribution is a probability distribution that is a generalization of the normal distribution. ...
In statistics, the Wishart distribution, named in honor of John Wishart, is any of a family of probability distributions for nonnegativedefinite matrixvalued random variables (random matrices). These distributions are of great importance in the estimation of covariance matrices in multivariate statistics. ...
Circular or directional statistics is the subdiscipline of statistics that deals with circular or directional data. ...
In mathematics, a degenerate distribution is the probability distribution of a random variable which always has the same value. ...
In probability, a singular distribution is a probability distribution concentrated on a measure zero set where the probability of each point in that set is zero. ...
Circular or directional statistics is the subdiscipline of statistics that deals with circular or directional data. ...
The 5parameter FisherBingham distribution or Kent distribution is a probability distribution on the threedimensional sphere. ...
In probability theory and statistics, the von Mises distribution is a continuous probability distribution. ...
In mathematics, a degenerate distribution is the probability distribution of a random variable which always has the same value. ...
In mathematics, a degenerate distribution is the probability distribution of a random variable which always has the same value. ...
The Dirac delta or Diracs delta, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function Î´(x) that has the value of infinity for x = 0 and the value zero elsewhere. ...
In probability, a singular distribution is a probability distribution concentrated on a measure zero set where the probability of each point in that set is zero. ...
The Cantor distribution is the probability distribution whose cumulative distribution function is the Cantor function. ...
In probability and statistics, an exponential family is any class of probability distributions having a certain form. ...
In probability theory, especially as that field is used in statistics, a locationscale family is a set of probability distributions on the real line parametrized by a location parameter Î¼ and a scale parameter Ïƒ â‰¥ 0; if X is any random variable whose probability distribution belongs to such a family, then...
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. ...
This article or section is incomplete and may require expansion and/or cleanup. ...
