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Information about Bimodal

“Bimodal” redirects here. For the musical concept, see Bimodality.


Figure 1. A simple bimodal distribution, in this case a mixture of two normal distributions with the same variance but different means. The figure shows the probability density function (p.d.f.), which is an average of the bell-shaped p.d.f.s of the two normal distributions.


In statistics, a bimodal distribution is a continuous probability distribution with two different modes. These appear as distinct peaks (local maxima) in the probability density function, as shown in Figure 1.

A good example is the height of a person. The heights of males form a roughly normal distribution, as do those of females. Each of these distributions is unimodal. However, if we plot a single histogram of the entire population, we see two peaks—one for males and one for females.

Bimodality is a property of many distributions. A bimodal distribution most commonly arises as a mixture of two different unimodal distributions. In other words, the bimodally distributed random variable X is defined as with probability or with probability , where Y and Z are unimodal random variables and is a mixture coefficient. In the height example, Y would be the height of a random male, Z the height of a random female, and the probability that a random individual is male.

Bimodal distributions are a commonly-used example of how summary statistics such as the mean, median, and standard deviation can be deceptive when used on an arbitrary distribution. For example, in the distribution in Figure 1, the mean and median would be about zero, even though zero is not a typical value. The standard deviation is also very large, even though the deviation of each normal distribution is relatively small.

More generally, a multimodal distribution is a continuous probability distribution with two or more modes, as illustrated in Figure 2. A unimodal distribution has only one mode.





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-MandelbrotEwens • 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' lambdaDirichlet • 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
Bimodality is the simultaneous use of two distinct pitch collections. It is more general than bitonality since the "scales" involved need not be traditional scales; if diatonic collections are involved, their pitch centers need not be the familiar major and minor-scale tonics.
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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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In probability theory, a probability distribution is called continuous if its cumulative distribution function is continuous. That is equivalent to saying that for random variables X with the distribution in question, Pr[X = a
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In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals.

Formally, a probability distribution has density f, if f
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normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. Each member of the family may be defined by two parameters, location and scale: the mean ("average",
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unimodal if for some value m (the mode), it is monotonically increasing for x ≤ m and monotonically decreasing for x ≥ m.
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In statistics, mean has two related meanings:
  • the arithmetic mean (and is distinguished from the geometric mean or harmonic mean).
  • the expected value of a random variable, which is also called the population mean.

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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 probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. It is usually denoted with the letter σ (lower case sigma).
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unimodal if for some value m (the mode), it is monotonically increasing for x ≤ m and monotonically decreasing for x ≥ m.
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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 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, ..., Xn) 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 Ni / N occupying a set of states i which each respectively possess energy Ei:


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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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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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Gauss-Kuzmin distribution gives the probability distribution of the occurrence of a given integer in the continued fraction expansion of an arbitrary real number. The distribution is named after Carl Friedrich Gauss, who first conjectured and studied the distribution around 1800,
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geometric distribution is either of two discrete probability distributions:
  • the probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set , or
  • the probability distribution of the number Y

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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.
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logarithmic distribution (also known as the logarithmic series distribution) is a discrete probability distribution derived from the Maclaurin series expansion



From this we obtain the identity


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negative binomial distribution is a discrete probability distribution. The Pascal distribution and the Polya distribution are special cases of the negative binomial.
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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. This can markedly improve the fit over a simple power-law relationship (see external link below).
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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.
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Rademacher distribution, named after Hans Rademacher is a discrete probability distribution which has a 50% chance for either 1 or -1. The probability mass function of this distribution is



The Rademacher distribution has been used in bootstrapping.
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Skellam distribution is the discrete probability distribution of the difference of two correlated or uncorrelated random variables and having Poisson distributions with different expected values and .
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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.
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Yule-Simon distribution is a discrete probability distribution named after Udny Yule and Herbert Simon. Simon originally called it the Yule distribution.

The probability mass function of the Yule-Simon(ρ) distribution is


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