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In probability theory and statistics, the beta prime distribution (also known as inverted beta distribution or beta distribution of the second kind [1]) is an absolutely continuous probability distribution. If [,] has a beta distribution, then the odds has a beta prime distribution.
In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] or (0, 1) in terms of two positive parameters, denoted by alpha (α) and beta (β), that appear as exponents of the variable and its complement to 1, respectively, and control the shape of the distribution.
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 .
The Beta distribution on [0,1], a family of two-parameter distributions with one mode, of which the uniform distribution is a special case, and which is useful in estimating success probabilities. The four-parameter Beta distribution, a straight-forward generalization of the Beta distribution to arbitrary bounded intervals [,].
Some distributions have been specially named as compounds: beta-binomial distribution, Beta negative binomial distribution, gamma-normal distribution. Examples: If X is a Binomial(n,p) random variable, and parameter p is a random variable with beta(α, β) distribution, then X is distributed as a Beta-Binomial(α,β,n).
The Type I cumulative distribution function is usually represented as a Poisson mixture of central beta random variables: [1] = = (+,),where λ is the noncentrality parameter, P(.) is the Poisson(λ/2) probability mass function, \alpha=m/2 and \beta=n/2 are shape parameters, and (,) is the incomplete beta function.
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The choice of the prior distribution is used to impose restrictions on , e.g. , with the beta distribution as a common choice due to (i) being defined between 0 and 1, (ii) being able to produce a variety of shapes, and (iii) yielding a posterior distribution of the standard form if combined with the likelihood function ().