Beta distribution

The beta distribution is a continuous probability distribution over the interval.

  • 2.1 extremum
  • 2.2 Expectation value
  • 2.3 variance
  • 2.4 standard deviation
  • 2.5 Coefficient of variation
  • 2.6 skewness
  • 2.7 symmetry
  • 3.1 Relationship to F-distribution
  • 3.2 Relationship to the gamma distribution

Definition

Beta distribution on [0,1]

Beta distribution is defined by the probability density

Outside of the interval it is continued by. It has the real parameter and. In order to guarantee their normalizability, is required.

The pre-factor is used to correct normalization. The term

Represents the Beta function, according to which the distribution is designated. This refers to the gamma function.

The distribution function is correspondingly

This function is also called regularized incomplete beta function.

Beta distribution on [a, b]

The general beta distribution is defined to

Wherein a and b are the lower and upper limits of the interval. Accordingly, the calculation of results to

The other models in this article applies only to the restricted to the interval Beta distribution.

Properties

Extremum

The density function takes its extremum at the position.

Expected value

The expected value is calculated as

Variance

The variance is given by

Standard deviation

For the standard deviation results

Coefficient of variation

From the expected value and variance are obtained directly the coefficient of variation

Skew

The skewness is given by

Symmetry

The beta distribution is symmetric about the skewness.

Relations with other distributions

Relationship to the F-distribution

When is F- distributed and then distributed

Relationship to the gamma distribution

If and independent gamma distributed random variables with parameters and, then the size is beta distributed with parameters and, shortly

Example

The beta distribution can be obtained from two gamma distributions: The quotient of the stochastically independent random variables, both of which are gamma distributed with parameters and respectively, is beta distributed with parameters and. and can be interpreted as a chi-square distribution with degrees of freedom respectively.

By means of linear regression, a regression line is placed by a point cloud with two pairs of values ​​, and statistical features, and such that the sum of squares of the vertical distances of the levels of the line are minimized.

The total scattering of y ( TSS) can be decomposed into the so-called scatter explained by a straight line estimated values ​​y * (ESS ) and the unexplained scatter of the residuals ( RSS ) with the scattering decomposition:

The coefficient of determination, the proportion of explained dispersion of the total scattering

Respectively

So is beta distributed. Since the coefficient of determination is the square of the correlation coefficient and also the square of the correlation coefficient is beta distributed.

However, the distribution of the coefficient of determination are specified in the model test of the regression through the F-distribution, which is present tabulated.

Beta-binomial distribution Boltzmann distribution Uniform distribution (discrete) Geometric distribution Logarithmic distribution Zeta distribution Triangular distribution Beta prime distribution Bose–Einstein statistics Erlang distribution Fermi–Dirac statistics Generalized extreme value distribution Hotelling's T-squared distribution Rayleigh distribution Weibull distribution Cauchy distribution Gumbel distribution Stable distribution Logistic distribution Multinomial distribution Dirichlet distribution Multivariate normal distribution Wishart distribution
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