Density, distribution function, quantile function and random generation
for the Birnbaum-Saunders distribution with shape parameter alpha,
scale parameter gamma, and location parameter mu.
Usage
dfatigue(x, alpha, gamma, mu = 0, log = FALSE)
pfatigue(q, alpha, gamma, mu = 0, lower.tail = TRUE, log.p = FALSE)
qfatigue(p, alpha, gamma, mu = 0, lower.tail = TRUE, log.p = FALSE)
rfatigue(n, alpha, gamma, mu = 0)Arguments
- x, q
vector of quantiles.
- alpha
shape parameter. Must be positive.
- gamma
scale parameter. Must be positive.
- mu
location parameter (default is 0).
- log, log.p
logical; if
TRUE, probabilities/densities p are returned as log(p).- lower.tail
logical; if
TRUE(default), probabilities are \(P[X \le x]\), 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.
Value
dfatigue gives the density, pfatigue gives the distribution function,
qfatigue gives the quantile function, and rfatigue generates random
deviates.
The length of the result is determined by n for rfatigue, and is the
maximum of the lengths of the numerical arguments for the other functions.
Details
The Birnbaum-Saunders distribution, also known as the fatigue life distribution, is commonly used in reliability and survival analysis. It was originally proposed to model fatigue failure times of materials subjected to cyclic stress.
The probability density function is given by: $$f(x) = \frac{\sqrt{(x-\mu)/\gamma} + \sqrt{\gamma/(x-\mu)}}{2\alpha(x-\mu)} \phi\left(\frac{\sqrt{(x-\mu)/\gamma} - \sqrt{\gamma/(x-\mu)}}{\alpha}\right)$$ for \(x > \mu\), where \(\phi\) is the standard normal density function.
References
Birnbaum, Z. W., & Saunders, S. C. (1969). A new family of life distributions. Journal of Applied Probability, 6(2), 319-327.
Examples
# Density at x = 2 with alpha = 0.5 and gamma = 1
dfatigue(2, alpha = 0.5, gamma = 1)
#> [1] 0.1556653
# CDF at x = 2
pfatigue(2, alpha = 0.5, gamma = 1)
#> [1] 0.9213504
# Quantile for p = 0.5 (median)
qfatigue(0.5, alpha = 0.5, gamma = 1)
#> [1] 1
# Generate 10 random values
rfatigue(10, alpha = 0.5, gamma = 1)
#> [1] 1.1360667 0.3155010 0.9972182 1.3627969 1.7622587 0.4139388 0.8837474
#> [8] 0.8851271 0.8682850 0.7588334