Beta Prime Distribution

Information about Beta Prime Distribution

Beta Prime
Probability density function
Cumulative distribution function
Parameters	$\text{[math]}$ shape (real) $\text{[math]}$ shape (real)
Support	$\text{[math]}$
Probability density function (pdf)	$\text{[math]}$
Cumulative distribution function (cdf)	$\text{[math]}$ where $\text{[math]}$ is the Gauss's hypergeometric function 2F1
Mean	$\text{[math]}$
Median
Mode	$\text{[math]}$
Variance	$\text{[math]}$
Skewness
Excess kurtosis
Entropy
Moment-generating function (mgf)
Characteristic function

A Beta Prime Distribution is a probability distribution defined for x>0 with two parameters (of positive real part), α and β, having the probability density function:

$\text{[math]}$

where $\text{[math]}$ is a Beta function. It is basically the same as the F distribution--if b is distributed as the beta prime distribution Beta'(α,β), then bβ/α obeys the F distribution with 2α and 2β degrees of freedom.

The mode of a variate $\text{[math]}$ distributed as $\text{[math]}$ is $\text{[math]}$ . Its mean is $\text{[math]}$ and its variance is $\text{[math]}$ .

If X is a $\text{[math]}$ variate then $\text{[math]}$ is a $\text{[math]}$ variate.

If X is a $\text{[math]}$ then $\text{[math]}$ and $\text{[math]}$ are $\text{[math]}$ and $\text{[math]}$ variates.

If X and Y are $\text{[math]}$ and $\text{[math]}$ variates, then $\text{[math]}$ is a $\text{[math]}$ variate.

References

MathWorld article

	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 probability theory and statistics, a shape parameter is a special kind of numerical parameter of a parametric family of probability distributions.

Definition

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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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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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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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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 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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beta function, also called the Euler integral of the first kind, is a special function defined by

for Re(x), Re(y) > 0.

The beta function was studied by Euler and Legendre and was given its name by Jacques Binet.
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F-distribution is a continuous probability distribution. It is also known as Snedecor's F distribution or the Fisher-Snedecor distribution (after R.A. Fisher and George W. Snedecor).
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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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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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Beta Prime Distribution

Information about Beta Prime Distribution

References

Definition

Introduction