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where I is the regularized incomplete beta function. While the related beta distribution is the conjugate prior distribution of the parameter of a Bernoulli distribution expressed as a probability, the beta prime distribution is the conjugate prior distribution of the parameter of a Bernoulli distribution expressed in odds.
The regularized incomplete beta function is the cumulative distribution function of the beta distribution, and is related to the cumulative distribution function (;,) of a random variable X following a binomial distribution with probability of single success p and number of Bernoulli trials n:
The beta function, , is a normalization constant to ensure that the total probability is 1. In the above equations is a ... The inverse of the harmonic mean ...
The quantile function, Q, of a probability distribution is the inverse of its cumulative distribution function F. The derivative of the quantile function, namely the quantile density function, is yet another way of prescribing a probability distribution. It is the reciprocal of the pdf composed with the quantile function.
The function A(t | ν) is the integral of Student's probability density function, f(t) between -t and t, for t ≥ 0 . It thus gives the probability that a value of t less than that calculated from observed data would occur by chance.
The beta family includes the beta of the first and second kind [7] (B1 and B2, where the B2 is also referred to as the Beta prime), which correspond to c = 0 and c = 1, respectively. Setting c = 0 {\displaystyle c=0} , b = 1 {\displaystyle b=1} yields the standard two-parameter beta distribution .
In mathematics, the inverse function of a function f (also called the inverse of f) is a function that undoes the operation of f. The inverse of f exists if and only if f is bijective , and if it exists, is denoted by f − 1 . {\displaystyle f^{-1}.}
Given a complex number z, there is not a unique complex number w satisfying erf w = z, so a true inverse function would be multivalued. However, for −1 < x < 1, ...