# distr-genbeta: The generalized Beta of the first kind Distribution In mbbefd: Maxwell Boltzmann Bose Einstein Fermi Dirac Distribution and Destruction Rate Modelling

## Description

Density, distribution function, quantile function and random generation for the GB1 distribution with parameters `shape0`, `shape1` and `shape2`.

## Usage

 ```1 2 3 4 5 6``` ```dgbeta(x, shape0, shape1, shape2, log = FALSE) pgbeta(q, shape0, shape1, shape2, lower.tail = TRUE, log.p = FALSE) qgbeta(p, shape0, shape1, shape2, lower.tail = TRUE, log.p = FALSE) rgbeta(n, shape0, shape1, shape2) ecgbeta(x, shape0, shape1, shape2) mgbeta(order, shape0, shape1, shape2) ```

## Arguments

 `x, q` vector of quantiles. `p` vector of probabilities. `n` number of observations. If `length(n) > 1`, the length is taken to be the number required. `shape0, shape1, shape2` positive parameters of the GB1 distribution. `log, log.p` logical; if TRUE, probabilities p are given as log(p). `lower.tail` logical; if TRUE (default), probabilities are P[X ≤ x], otherwise, P[X > x]. `order` order of the raw moment.

## Details

The GB1 distribution with parameters `shape0` = g, `shape1` = a and `shape2` = b has density

Γ(a+b)/(Γ(a)Γ(b))x^(a/g-1)(1-x^{1/g})^(b-1)/g

for a,b,g > 0 and 0 ≤ x ≤ 1 where the boundary values at x=0 or x=1 are defined as by continuity (as limits).

## Value

`dgbeta` gives the density, `pgbeta` the distribution function, `qgbeta` the quantile function, and `rgbeta` generates random deviates.

## References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.

Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions. New York: Dover. Chapter 6: Gamma and Related Functions.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume 2, especially chapter 25. Wiley, New York.

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