NationMaster - Encyclopedia: Binomial distribution

Binomial
Probability mass function The lines connecting the dots are added for clarity
Cumulative distribution function Colors match the image above
Parameters	$n geq 0$ number of trials (integer) $0leq p leq 1$ success probability (real)
Support	$k in {0,dots,n}!$
Probability mass function (pmf)	${nchoose k} p^k (1-p)^{n-k} !$
Cumulative distribution function (cdf)	$I_{1-p}(n-lfloor krfloor, 1+lfloor krfloor) !$
Mean	$np!$
Median	one of ${lfloor nprfloor-1, lfloor nprfloor, lfloor nprfloor+1}$
Mode	$lfloor (n+1),prfloor!$
Variance	$np(1-p)!$
Skewness	$frac{1-2p}{sqrt{np(1-p)}}!$
Excess kurtosis	$frac{1-6p(1-p)}{np(1-p)}!$
Entropy	$frac{1}{2} ln left( 2 pi n e p (1-p) right) + O left( frac{1}{n} right)$
Moment-generating function (mgf)	$(1-p + pe^t)^n !$
Characteristic function	$(1-p + pe^{it})^n !$

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. Such a success/failure experiment is also called a Bernoulli experiment or Bernoulli trial. In fact, when n = 1, then the binomial distribution is the Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... The integers are commonly denoted by the above symbol. ... 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 real-valued 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 real-valued random variable, X. For every real number x, the cdf is given by where the right-hand 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... In probability theory and statistics, a median is a number dividing the higher half of a sample, a population, or a probability distribution from the lower half. ... In, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. ... In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ... Example of the experimental data with non-zero 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 real-valued random variable. ... In probability theory and statistics, kurtosis is a measure of the peakedness of the probability distribution of a real-valued random variable. ... Entropy of a Bernoulli trial as a function of success probability, often called the binary entropy function In information theory, information entropy or Shannons entropy is a measure of the average information content associated with the outcome of a random variable. ... In probability theory and statistics, the moment-generating 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. ... It has been suggested that this article or section be merged with Probability axioms. ... Template:Otherusescccc A graph of a bell curve in a normal distribution showing statistics used in educational assessment, comparing various grading methods. ... In mathematics, a probability distribution is called discrete, if it is fully characterized by a probability mass function. ... Probability is the extent to which something is likely to happen or be the case[1]. Probability theory is used extensively in areas such as statistics, mathematics, science, philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems. ... 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. ... 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 statistics, the binomial test is an exact test of the statistical significance of deviations from a theoretically expected distribution of observations into two categories. ... In statistics, a result is significant if it is unlikely to have occurred by chance, given that a presumed null hypothesis is true. ...

1 Examples
2 Specification
- 2.1 Probability mass function
- 2.2 Cumulative distribution function
3 Mean, standard deviation, and mode
4 Relatonship to other distributions
5 Limits of binomial distributions
6 References
7 See also
8 External links

Examples

An elementary example is this: roll a die ten times and count the number of 1s as outcome. Then this random number follows a binomial distribution with n = 10 and p = 1/6.

A typical example is the following: assume 5% of the population is green-eyed. You pick 500 people randomly. The number of green-eyed people you pick is a random variable X which follows a binomial distribution with n = 500 and p = 0.05 (when picking the people with replacement). A random variable is a mathematical function that maps outcomes of random experiments to numbers. ...

Specification

Probability mass function

In general, if the random variable K follows the binomial distribution with parameters n and p, we write k ~ B(n, p). The probability of getting exactly k successes is given by the probability mass function: In probability theory, a probability mass function (abbreviated pmf) gives the probability that a discrete random variable is exactly equal to some value. ...

$f(k;n,p)={nchoose k}p^k(1-p)^{n-k}$

for k = 0, 1, 2, ..., n and where

${nchoose k}=frac{n!}{k!(n-k)!}$

is the binomial coefficient (hence the name of the distribution) "n choose k" (also denoted C(n, k) or nCk). The formula can be understood as follows: we want k successes (p^k) and n − k failures (1 − p)^{n − k}. However, the k successes can occur anywhere among the n trials, and there are C(n, k) different ways of distributing k successes in a sequence of n trials. In mathematics, particularly in combinatorics, the binomial coefficient of the natural number n and the integer k is the number of combinations that exist. ...

In creating reference tables for binomial distribution probability, usually the table is filled in up to n/2 values. This is because for k > n/2, the probability can be calculated by its complement as

$f(k;n,p)=f(n-k;n,1-p).,!$

So, one must look to a different k and a different p (the binomial is not symmetrical in general).

Cumulative distribution function

The cumulative distribution function can be expressed in terms of the regularized incomplete beta function, as follows: In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than... In mathematics, the incomplete beta function is a generalization of the beta function that replaces the definite integral of the beta function with an indefinite integral. ...

$F(k;n,p) = Pr(X le k) = I_{1-p}(n-k, k+1) !$

provided k is an integer and 0 ≤ k ≤ n. If x is not necessarily an integer or not necessarily positive, one can express it thus:

$F(x;n,p) = Pr(X le x) = sum_{j=0}^{operatorname{Floor}(x)} {nchoose j}p^j(1-p)^{n-j}$

For k ≤ np, upper bounds for the lower tail of the distribution function can be derived. In particular, Hoeffding's inequality yields the bound In probability theory, the Chernoff bound, named after Herman Chernoff, gives a lower bound for the success of majority agreement for n independent, equally likely events. ... Hoeffdings inequality, named after Wassily Hoeffding, is a result in probability theory that gives an upper bound on the probability for the sum of random variables to deviate from its expected value. ...

$F(k;n,p) leq expleft(-2 frac{(np-k)^2}{n}right), !$

and Chernoff's inequality can be used to derive the bound In probability theory, Chernoffs inequality, named after Herman Chernoff, states the following. ...

$F(k;n,p) leq expleft(-frac{1}{2,p} frac{(np-k)^2}{n}right). !$

Mean, standard deviation, and mode

If X ~ B(n, p) (that is, X is a binomially distributed random variable), then the expected value of X is 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...

$operatorname{E}(X)=np,!$

and the variance is In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ...

$operatorname{Var}(X)=np(1-p).,!$

This fact is easily proven as follows. Suppose first that we have exactly one Bernoulli trial. We have two possible outcomes, 1 and 0, with the first having probability p and the second having probability 1 − p; the mean for this trial is given by μ = p. Using the definition of variance, we have In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ...

$sigma^2= left(1 - pright)^2p + (0-p)^2(1 - p) = p(1-p).$

Now suppose that we want the variance for n such trials (i.e. for the general binomial distribution). Since the trials are independent, we may add the variances for each trial, giving

$sigma^2_n = sum_{k=1}^n sigma^2 = np(1 - p). quad$

The most likely value or mode of X is given by the largest integer less than or equal to (n + 1)p; if m = (n + 1)p is itself an integer, then m − 1 and m are both modes. In, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. ...

Relatonship to other distributions

Kitchen's Theorem

Given the fact that we can define the Binomial and Negative Binomial, we can say by Kicthen's Theorem that there is a relationship between the distributions can be defined

P[X>r] for the Binomial = P[r<X<n] for negative binomial

Binomial

$P[X=r]={nchoose r}p^r(1-p)^{n-r}$

for r = 0, 1, 2, ..., n and where

${nchoose r}=frac{n!}{r!(n-r)!}$

Negative Binomial

$P[X=x] = {x-1choose r-1}; p^r , (1-p)^k !$

where k in the -ve binomial = x-r

where r is the number of successes in the Binomial, and n is the number of trials taken. However, in the Negative the same parameters, represent r = exact number of successes needed, and n is the is the number of trials needed to get the desired amount of successes.

In other words Kitchens Theorem states

P[of getting more than r successes in exactly n trials] = P[of it taking between r and n trials to get exactly r successes]

Sums of binomials

If X ~ B(n, p) and Y ~ B(m, p) are independent binomial variables, then X + Y is again a binomial variable; its distribution is

$X+Y sim B(n+m, p).,$

Normal approximation

Binomial PDF and normal approximation for n = 6 and p = 0.5.

If n is large enough, the skew of the distribution is not too great, and a suitable continuity correction is used, then an excellent approximation to B(n, p) is given by the normal distribution Binomial PDF (n=6,p=0. ... Binomial PDF (n=6,p=0. ... In probability theory, if a random variable X has a binomial distribution with parameters n and p, i. ... The normal distribution, also called Gaussian distribution by scientists (named after Carl Friedrich Gauss due to his rigorous application of the distribution to astronomical data (Havil, 2003)) is a probability distribution of great importance in many fields. ...

$operatorname{N}(np, np(1-p)).,!$

Various rules of thumb may be used to decide whether n is large enough. One rule is that both np and n(1 − p) must be greater than 5. However, the specific number varies from source to source, and depends on how good an approximation one wants; some sources give 10. Another commonly used rule holds that the above normal approximation is appropriate only if A rule of thumb is an easily learned and easily applied procedure for approximately calculating or recalling some value, or for making some determination. ...

$mu pm 3 sigma = np pm 3 sqrt{np(1-p)} in [0,n].$

The following is an example of applying a continuity correction: Suppose one wishes to calculate Pr(X ≤ 8) for a binomial random variable X. If Y has a distribution given by the normal approximation, then Pr(X ≤ 8) is approximated by Pr(Y ≤ 8.5). The addition of 0.5 is the continuity correction. Warning: The normal approximation gives inaccurate results unless a continuity correction is used. In probability theory, if a random variable X has a binomial distribution with parameters n and p, i. ...

This approximation is a huge time-saver (exact calculations with large n are very onerous); historically, it was the first use of the normal distribution, introduced in Abraham de Moivre's book The Doctrine of Chances in 1733. Nowadays, it can be seen as a consequence of the central limit theorem since B(n, p) is a sum of n independent, identically distributed 0-1 indicator variables. Abraham de Moivre. ... The Doctrine of Chances is a book on probability theory by 18th-century French mathematician Abraham de Moivre, published in 1733. ... A central limit theorem is any of a set of weak-convergence results in probability theory. ... In the mathematical subfield of set theory, the indicator function is a function defined on a set X which is used to indicate membership of an element in a subset A of X. Remark. ...

For example, suppose you randomly sample n people out of a large population and ask them whether they agree with a certain statement. The proportion of people who agree will of course depend on the sample. If you sampled groups of n people repeatedly and truly randomly, the proportions would follow an approximate normal distribution with mean equal to the true proportion p of agreement in the population and with standard deviation σ = (p(1 − p)/n)^1/2. Large sample sizes n are good because the standard deviation gets smaller, which allows a more precise estimate of the unknown parameter p. Sample size, usually designated N, is the number of repeated measurements in a statistical sample. ...

Poisson approximation

The binomial distribution converges towards the Poisson distribution as the number of trials goes to infinity while the product np remains fixed. Therefore the Poisson distribution with parameter λ = np can be used as an approximation to B(n, p) of the binomial distribution if n is sufficiently large and p is sufficiently small. According to one rule of thumb, this approximation is good if n ≥ 20 and p ≤ 0.05, and also if n ≥ 100 and np ≤ 10.^[1] In probability theory and statistics, the Poisson distribution is a discrete probability distribution. ...

Limits of binomial distributions

As n approaches ∞ and p approaches 0 while np remains fixed at λ > 0 or at least np approaches λ > 0, then the Binomial(n, p) distribution approaches the Poisson distribution with expected value λ.

As n approaches ∞ while p remains fixed, the distribution of

${X-np over sqrt{np(1-p) }}$

approaches the normal distribution with expected value 0 and variance 1.

In probability theory and statistics, the Poisson distribution is a discrete probability distribution. ... 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... The normal distribution, also called Gaussian distribution by scientists (named after Carl Friedrich Gauss due to his rigorous application of the distribution to astronomical data (Havil, 2003)) is a probability distribution of great importance in many fields. ... In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ...

References

^ NIST/SEMATECH, '6.3.3.1. Counts Control Charts', e-Handbook of Statistical Methods, <http://www.itl.nist.gov/div898/handbook/pmc/section3/pmc331.htm> [accessed 25 October 2006]

Abdi, H. "[1] ((2007). Binomial Distribution: Binomial and Sign Tests.. In N.J. Salkind (Ed.): Encyclopedia of Measurement and Statistics. Thousand Oaks (CA): Sage.".

Luc Devroye, Non-Uniform Random Variate Generation, New York: Springer-Verlag, 1986. See especially Chapter X, Discrete Univariate Distributions.

Voratas Kachitvichyanukul and Bruce W. Schmeiser, Binomial random variate generation, Communications of the ACM 31(2):216–222, February 1988. DOI:10.1145/42372.42381

Cheatam & Steele, "Uniform Distributive Norms", Los Angeles: Time-Warner, 1998.

Communications of the ACM (CACM) is the flagship monthly magazine of the Association for Computing Machinery. ... A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ...

	Probability distributions [ view • talk • edit ]
	Univariate	Multivariate
Discrete:	Benford • Bernoulli • binomial • Boltzmann • categorical • compound Poisson • 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 • Erlang • exponential • exponential power • F • fading • 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 • inverse Gaussian • inverse gamma • Kumaraswamy • Landau • Laplace • Lévy • Lévy skew alpha-stable • logistic • log-normal • Maxwell-Boltzmann • Maxwell speed • normal (Gaussian) • normal inverse Gaussian • Pareto • Pearson • polar • raised cosine • Rayleigh • relativistic Breit-Wigner • Rice • shifted Gompertz • Student's t • triangular • type-1 Gumbel • type-2 Gumbel • uniform • Variance-Gamma • Voigt • von Mises • Weibull • Wigner semicircle • Wilks' lambda	Dirichlet • Kent • matrix normal • multivariate normal • multivariate Student • von Mises-Fisher • Wigner quasi • Wishart
Miscellaneous:	Cantor • conditional • exponential family • infinitely divisible • location-scale family • marginal • maximum entropy • phase-type • posterior • prior • quasi • sampling • singular

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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 semi-ellipse... This article or section is in need of attention from an expert on the subject. ... 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). ... The 5-parameter Fisher-Bingham distribution or Kent distribution is a probability distribution on the three-dimensional sphere. ... The matrix normal distribution is a probability distribution that is a generalization of the normal distribution. ... 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 one-dimensional normal distribution (also called a Gaussian distribution). ... In statistics, a multivariate Student distribution is a multivariate generalization of the Students t-distribution. ... Points sampled from three von Mises-Fisher distributions on the sphere (blue: , green: , red: ). The mean directions are shown with arrows. ... The Wigner quasi-probability distribution was introduced by Eugene Wigner in 1932 to study quantum corrections to classical statistical mechanics. ... In statistics, the Wishart distribution, named in honor of John Wishart, is any of a family of probability distributions for nonnegative-definite matrix-valued random variables (random matrices). These distributions are of great importance in the estimation of covariance matrices in multivariate statistics. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... The Cantor distribution is the probability distribution whose cumulative distribution function is the Cantor function. ... This article defines some terms which characterize probability distributions of two or more variables. ... In probability and statistics, an exponential family is any class of probability distributions having a certain form. ... 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 theory, especially as that field is used in statistics, a location-scale 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 probability theory, given two jointly distributed random variables X and Y, the marginal distribution of X is simply the probability distribution of X ignoring information about Y, typically calculated by summing or integrating the joint probability distribution over Y. For discrete random variables, the marginal probability mass function can... 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. ... A phase-type distribution is a probability distribution that results from a system one or more inter-related poisson processes occurring in sequence, or phases. ... The posterior probability can be calculated by Bayes theorem from the prior probability and the likelihood function. ... A prior probability is a marginal probability, interpreted as a description of what is known about a variable in the absence of some evidence. ... // Introduction In the most general form, the dynamics of a quantum mechanical system are determined by a master equation - an equation of motion for the density operator (usually written ) of the system. ... In statistics, a sampling distribution is the probability distribution, under repeated sampling of the population, of a given statistic (a numerical quantity calculated from the data values in a sample). ... 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. ...

External links

Binomial Probability Distribution Calculator
Binomial Probabilities Simple Explanation
SOCR Binomial Distribution Applet

Categories: Discrete distributions | Factorial and binomial topics | Probability and statistics

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