Uniform Distribution (discrete)

Information about Uniform Distribution (discrete)

discrete uniform
Probability mass function Discrete uniform probability mass function for n=5 n=5 where n=b-a+1
Cumulative distribution function Discrete uniform cumulative density function for n=5
Parameters	$\text{[math]}$ $\text{[math]}$ $\text{[math]}$
Support	$\text{[math]}$
Probability mass function (pmf)	$\text{[math]}$
Cumulative distribution function (cdf)	$\text{[math]}$
Mean	$\text{[math]}$
Median	$\text{[math]}$
Mode	N/A
Variance	$\text{[math]}$
Skewness	$\text{[math]}$
Excess kurtosis	$\text{[math]}$
Entropy	$\text{[math]}$
Moment-generating function (mgf)	$\text{[math]}$
Characteristic function	$\text{[math]}$

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.

If a random variable has any of $\text{[math]}$ possible values $\text{[math]}$ that are equally probable, then it has a discrete uniform distribution. The probability of any outcome $\text{[math]}$ is $\text{[math]}$ . A simple example of the discrete uniform distribution is throwing a fair die. The possible values of $\text{[math]}$ are 1, 2, 3, 4, 5, 6; and each time the die is thrown, the probability of a given score is 1/6.

In case the values of a random variable with a discrete uniform distribution are real, it is possible to express the cumulative distribution function in terms of the degenerate distribution; thus

$\text{[math]}$

where the Heaviside step function $\text{[math]}$ is the CDF of the degenerate distribution centered at $\text{[math]}$ . This assumes that consistent conventions are used at the transition points.

See rencontres numbers for an account of the probability distribution of the number of fixed points of a uniformly distributed random permutation.

	Probability distributions [ • • [ edit] ]
	Univariate	Multivariate
Discrete:	Benford • Bernoulli • binomial • Boltzmann • categorical • compound Poisson • discrete phase-type • degenerate • Gauss-Kuzmin • geometric • hypergeometric • logarithmic • negative binomial • parabolic fractal • Poisson • Rademacher • Skellam • uniform • Yule-Simon • zeta • Zipf • Zipf-Mandelbrot	Ewens • multinomial • multivariate Polya
Continuous:	Beta • Beta prime • Cauchy • chi-square • Dirac delta function • Coxian • Erlang • exponential • exponential power • F • fading • Fermi-Dirac • Fisher's z • Fisher-Tippett • Gamma • generalized extreme value • generalized hyperbolic • generalized inverse Gaussian • Half-Logistic • Hotelling's T-square • hyperbolic secant • hyper-exponential • hypoexponential • inverse chi-square (scaled inverse chi-square) • inverse Gaussian • inverse gamma (scaled inverse gamma) • Kumaraswamy • Landau • Laplace • Lvy • Lvy skew alpha-stable • logistic • log-normal • Maxwell-Boltzmann • Maxwell speed • Nakagami • normal (Gaussian) • normal-gamma • normal inverse Gaussian • Pareto • Pearson • phase-type • polar • raised cosine • Rayleigh • relativistic Breit-Wigner • Rice • shifted Gompertz • Student's t • triangular • truncated normal • type-1 Gumbel • type-2 Gumbel • uniform • Variance-Gamma • Voigt • von Mises • Weibull • Wigner semicircle • Wilks' lambda	Dirichlet • Generalized Dirichlet distribution . inverse-Wishart • Kent • matrix normal • multivariate normal • multivariate Student • von Mises-Fisher • Wigner quasi • Wishart
Miscellaneous:	bimodal • Cantor • conditional • equilibrium • exponential family • infinitely divisible • location-scale family • marginal • maximum entropy • posterior • prior • quasi • sampling • singular

In mathematics, a support of a function f from a set X to the real numbers R is a subset Y of X such that f (x) is zero for all x in X and outside Y.
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probability mass function (abbreviated pmf) is a function that gives the probability that a discrete random variable is exactly equal to some value. A probability mass function differs from a probability density function (abbreviated pdf
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In probability theory, the cumulative distribution function (CDF), also called probability distribution function or just distribution function,^[1] completely describes the probability distribution of a real-valued random variable X.
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expected value (or mathematical expectation, or mean) of a discrete random variable is the sum of the probability of each possible outcome of the experiment multiplied by the outcome value (or payoff).
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median is described as the number separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to highest value and picking
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In statistics, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. The term is applied both to probability distributions and to collections of experimental data.
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variance of a random variable (or somewhat more precisely, of a probability distribution) is one measure of statistical dispersion, averaging the squared distance of its possible values from the expected value.
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skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable.

Introduction

Consider the distribution in the figure. The bars on the right side of the distribution taper differently than the bars on the left side.
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kurtosis (from the Greek word kurtos, meaning bulging) is a measure of the "peakedness" of the probability distribution of a real-valued random variable. Higher kurtosis means more of the variance is due to infrequent extreme deviations, as opposed to frequent
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Shannon entropy or information entropy is a measure of the uncertainty associated with a random variable.

Shannon entropy quantifies the information contained in a piece of data: it is the minimum average message length, in bits (if using base-2 logarithms), that must
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In probability theory and statistics, the moment-generating function of a random variable X is

wherever this expectation exists. The moment-generating function generates the moments of the probability distribution.
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In probability theory, the characteristic function of any random variable completely defines its probability distribution. On the real line it is given by the following formula, where X is any random variable with the distribution in question:

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Probability theory is the branch of mathematics concerned with analysis of random phenomena.^[1] The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities
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Statistics is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. It is applicable to a wide variety of academic disciplines, from the physical and social sciences to the humanities.
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probability distribution that assigns a probability to every subset (more precisely every measurable subset) of its state space in such a way that the probability axioms are satisfied.
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In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and
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degenerate distribution is the probability distribution of a discrete random variable whose support consists of only one value. Examples include a two-headed coin and rolling a die whose sides all show the same number.
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step function if it can be written as a finite linear combination of indicator functions of half-open intervals. Informally speaking, a step function is a piecewise constant function having only finitely many pieces.
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In combinatorial mathematics, the rencontres numbers are a triangular array of integers that enumerate permutations of the set with specified numbers of fixed points. (Rencontre is French for encounter. By some accounts, the problem is named after a solitaire game.
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A random permutation is a random ordering of a set of objects, that is, a permutation-valued random variable. The use of random permutations is often fundamental to fields that use randomized algorithms. Such fields include coding theory, cryptography, and simulation.
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In statistics, in univariate data, each data point has only one scalar component. Or, when the statistical technique to be used, it contains only one dependent variable. The more general case is multivariate.
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A multivariate random variable or random vector is a vector X = (X₁, ..., X_n) whose components are scalar-valued random variables on the same probability space (Ω, P).
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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 .
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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.
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Boltzmann distribution predicts the distribution function for the fractional number of particles N_i / N occupying a set of states i which each respectively possess energy E_i:

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A categorical distribution is the most general distribution whose sample space is the set .

It is the generalization of the Bernoulli distribution for a categorical random variable.

It should not be confused with the multinomial distribution.
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In probability theory, a compound Poisson distribution is the probability distribution of a "Poisson-distributed number" of independent identically-distributed random variables. More precisely, suppose

i.e.
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Uniform Distribution (discrete)

Information about Uniform Distribution (discrete)

Introduction