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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.
The formulation of the beta distribution discussed here is also known as the beta distribution of the first kind, whereas beta distribution of the second kind is an alternative name for the beta prime distribution. The generalization to multiple variables is called a Dirichlet 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 =, = 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 [,].
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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(< +),. which is a Gumbel distribution. A simple proof by martingales is in the next section . Donald J. Newman and Lawrence Shepp gave a generalization of the coupon collector's problem when m copies of each coupon need to be collected.