# DiscreteWeibull: Discrete Weibull distribution (type I) In extraDistr: Additional Univariate and Multivariate Distributions

## Description

Density, distribution function, quantile function and random generation for the discrete Weibull (type I) distribution.

## Usage

 ```1 2 3 4 5 6 7``` ```ddweibull(x, shape1, shape2, log = FALSE) pdweibull(q, shape1, shape2, lower.tail = TRUE, log.p = FALSE) qdweibull(p, shape1, shape2, lower.tail = TRUE, log.p = FALSE) rdweibull(n, shape1, shape2) ```

## Arguments

 `x, q` vector of quantiles. `shape1, shape2` parameters (named q, β). Values of `shape2` need to be positive and `0 < shape1 < 1`. `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]. `p` vector of probabilities. `n` number of observations. If `length(n) > 1`, the length is taken to be the number required.

## Details

Probability mass function

f(x) = q^x^β - q^(x+1)^β

Cumulative distribution function

F(x) = 1-q^(x+1)^β

Quantile function

F^-1(p) = ceiling((log(1-p)/log(q))^(1/β) - 1)

## References

Nakagawa, T. and Osaki, S. (1975). The Discrete Weibull Distribution. IEEE Transactions on Reliability, R-24, 300-301.

Kulasekera, K.B. (1994). Approximate MLE's of the parameters of a discrete Weibull distribution with type I censored data. Microelectronics Reliability, 34(7), 1185-1188.

Khan, M.A., Khalique, A. and Abouammoh, A.M. (1989). On estimating parameters in a discrete Weibull distribution. IEEE Transactions on Reliability, 38(3), 348-350.

`Weibull`
 ``` 1 2 3 4 5 6 7 8 9 10``` ```x <- rdweibull(1e5, 0.32, 1) xx <- seq(-2, 100, by = 1) plot(prop.table(table(x)), type = "h") lines(xx, ddweibull(xx, .32, 1), col = "red") # Notice: distribution of F(X) is far from uniform: hist(pdweibull(x, .32, 1), 50) plot(ecdf(x)) lines(xx, pdweibull(xx, .32, 1), col = "red", lwd = 2, type = "s") ```