BETA FUNCTION PDF
PDF | The aim of this paper is to study gamma and beta functions of complex variable. Further, we prove some properties of gamma and beta. integer values, the Gamma function was later presented in its traditional integral form by and the regularized (normalized) form of the incomplete Beta function. Beta Function and its Applications Riddhi D. ~ Department of Physics and Astronomy The University of Tennessee Knoxville, TN , USA Abstract m!n!.
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Gamma and Beta. Functions. The Gamma Function eo. The gamma function may be regarded as a generalization of n! (n-factorial), where n is any positive. The gamma and the beta function. As mentioned in the book , see page 6, the integral representation () is often taken as a definition for the gamma. Abstract—In this paper, the partial derivatives Bp,q(x, y) = ∂q+p. ∂xp∂yq B(x, y) of the Beta function B(x, y) are expressed in terms of a finite.
Skip to main content. Log In Sign Up. Beta Function and its Applications. Beta Function and its Applications Riddhi D. Rewriting the arguments then gives the usual form The Beta function was …rst studied by Euler and for the beta function, Legendre and was given its name by Jacques Binet. It also occurs  in the theory of the preferential attachment process, a type of stochastic urn process.
As Mosteller and Tukey remark  p. This illustrates how, for short-tailed distributions, the extreme observations should get more weight. The logarithm of the geometric mean G X of a distribution with random variable X is the arithmetic mean of ln X , or, equivalently, its expected value:.
Numerical values for the relative error in this approximation follow: The fundamental property of the geometric mean, which can be proven to be false for any other mean, is. This makes the geometric mean the only correct mean when averaging normalized results, that is results that are presented as ratios to reference values.
The geometric mean plays a central role in maximum likelihood estimation, see section "Parameter estimation, maximum likelihood. This equality follows from the following symmetry displayed between both geometric means:.
The harmonic mean plays a role in maximum likelihood estimation for the four parameter case, in addition to the geometric mean. Actually, when performing maximum likelihood estimation for the four parameter case, besides the harmonic mean H X based on the random variable X , also another harmonic mean appears naturally: This equality follows from the following symmetry displayed between both harmonic means:.
Also, the following limits with only the noted variable approaching the limit can be obtained from the above expressions:. The logarithm of the geometric variance, ln var GX , of a distribution with random variable X is the second moment of the logarithm of X centered on the geometric mean of X , ln G X:. For a beta distribution, higher order logarithmic moments can be derived by using the representation of a beta distribution as a proportion of two Gamma distributions and differentiating through the integral.
They can be expressed in terms of higher order poly-gamma functions. See the section titled "Other moments, Moments of transformed random variables, Moments of logarithmically transformed random variables". The log geometric variances are positive for all values of the shape parameters. This equality follows from the following symmetry displayed between both log geometric variances:.
Beta distribution - Wikipedia
Therefore, the effect of very large deviations from the mean are not as overly weighted. Using Stirling's approximation to the Gamma function , N. The mean absolute difference for the Beta distribution is:. The Gini coefficient for the Beta distribution is half of the relative mean absolute difference:. The beta distribution has been applied in acoustic analysis to assess damage to gears, as the kurtosis of the beta distribution has been reported to be a good indicator of the condition of a gear.
As persons or other targets moving on the ground generate continuous signals in the form of seismic waves, one can separate different targets based on the seismic waves they generate. Kurtosis is sensitive to impulsive signals, so it's much more sensitive to the signal generated by human footsteps than other signals generated by vehicles, winds, noise, etc.
To prevent confusion  between kurtosis the fourth moment centered on the mean, normalized by the square of the variance and excess kurtosis, when using symbols, they will be spelled out as follows: The description of kurtosis as a measure of the "potential outliers" or "potential rare, extreme values" of the probability distribution, is correct for all distributions including the beta distribution. When rare, extreme values can occur in the beta distribution, the higher its kurtosis; otherwise, the kurtosis is lower.
Minimum kurtosis takes place when the mass density is concentrated equally at each end and therefore the mean is at the center , and there is no probability mass density in between the ends.
The excess kurtosis can also be expressed in terms of just the following two parameters: A coin toss: Variance is maximum because the distribution is bimodal with nothing in between the two modes spikes at each end. Excess kurtosis is minimum: Excess kurtosis reaches the minimum possible value for any distribution when the probability density function has two spikes at each end: Alternatively, the excess kurtosis can also be expressed in terms of just the following two parameters: From this last expression, one can obtain the same limits published practically a century ago by Karl Pearson in his paper,  for the beta distribution see section below titled "Kurtosis bounded by the square of the skewness".
The characteristic function is the Fourier transform of the probability density function. The characteristic function of the beta distribution is Kummer's confluent hypergeometric function of the first kind: Also, the real and imaginary parts of the characteristic function enjoy the following symmetries with respect to the origin of variable t:.
It also follows   that the moment generating function is. Using the moment generating function , the k -th raw moment is given by  the factor. It can also be written in a recursive form as.
These are the expected values for inverted variables, these are related to the harmonic means, see section titled "Harmonic mean":. Variances of these transformed variables can be obtained by integration, as the expected values of the second moments centered on the corresponding variables:.
These expectations and variances appear in the four-parameter Fisher information matrix section titled "Fisher information," "four parameters". Expected values for logarithmic transformations useful for maximum likelihood estimates, see section titled "Parameter estimation, Maximum likelihood" below are discussed in this section.
Logit transformations are interesting,  as they usually transform various shapes including J-shapes into usually skewed bell-shaped densities over the logit variable, and they may remove the end singularities over the original variable:.
Higher order logarithmic moments can be derived by using the representation of a beta distribution as a proportion of two Gamma distributions and differentiating through the integral. They can be expressed in terms of higher order poly-gamma functions as follows:. These logarithmic variances and covariance are the elements of the Fisher information matrix for the beta distribution. They are also a measure of the curvature of the log likelihood function see section on Maximum likelihood estimation.
It also follows that the variances of the logit transformed variables are:. It is to be expected that the maximum entropy should take place when the beta distribution becomes equal to the uniform distribution, since uncertainty is maximal when all possible events are equiprobable.
The continuous case differential entropy was introduced by Shannon in his original paper where he named it the "entropy of a continuous distribution" , as the concluding part  of the same paper where he defined the discrete entropy. It is known since then that the differential entropy may differ from the infinitesimal limit of the discrete entropy by an infinite offset, therefore the differential entropy can be negative as it is for the beta distribution.
What really matters is the relative value of entropy. The cross entropy has been used as an error metric to measure the distance between two hypotheses. It is the information measure most closely related to the log maximum likelihood  see section on "Parameter estimation. Maximum likelihood estimation". It is defined as follows measured in nats.
The relative entropy, or Kullback—Leibler divergence , is always non-negative. A few numerical examples follow:. The value of the Kullback divergence depends on the direction traveled: In the numerical example above, the Kullback divergence measures the inefficiency of assuming that the distribution is bell-shaped Beta 3, 3 , rather than uniform Beta 1, 1.
The situation is analogous to the incomplete gamma function being a generalization of the gamma function. To derive the integral representation of the beta function, we 3-D Image of Beta Function R 1write the product R1 of two factorial as m! A somewhat more straightforward derivation: Again, now the comparison to x; y leads us to: This leads to an easy identi…cation with the ex- is the gamma function p.
The second identity shows pected result: The Euler Beta function appeared in elementary particle physics as a model for the scattering ampli- tude in the so-called "dual resonance model".
In- troduced by Veneziano in the th in order to …t experimental data , it soon turned out that the basic physics behind this model is the string instead of the zero-dimensional mass-point.
The principal reason for scienti…c interest in preferential attachment is that it can, under suitable circumstances, generate power law distributions of wealth. A preferential at- tachment process is an urn process in which addi- Theory: In Veneziano ,an Italian theoretical physicist and a the most commonly studied examples, the number of founder of string theory.
In the most general form and Leonard Susskind of Stanford University revealed of the process, balls are added to the system at an that the nuclear interactions of elementary particles overall rate of m new species for each new urn.
Probability density Function References: Zelen and N. Severo in Milton Abramowitz and Irene A. Stegun, eds. Or- lando, FL: Academic Press, pp.
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