extraDistr (version 1.8.1)

Rayleigh: Rayleigh distribution

Description

Density, distribution function, quantile function and random generation for the Rayleigh distribution.

Usage

drayleigh(x, sigma = 1, log = FALSE)
prayleigh(q, sigma = 1, lower.tail = TRUE, log.p = FALSE)
qrayleigh(p, sigma = 1, lower.tail = TRUE, log.p = FALSE)
rrayleigh(n, sigma = 1)

Arguments

x, q

vector of quantiles.

sigma

positive valued parameter.

log, log.p

logical; if TRUE, probabilities p are given 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.

Details

Probability density function $$ f(x) = \frac{x}{\sigma^2} \exp\left(-\frac{x^2}{2\sigma^2}\right) $$

Cumulative distribution function $$ F(x) = 1 - \exp\left(-\frac{x^2}{2\sigma^2}\right) $$

Quantile function $$ F^{-1}(p) = \sqrt{-2\sigma^2 \log(1-p)} $$

References

Krishnamoorthy, K. (2006). Handbook of Statistical Distributions with Applications. Chapman & Hall/CRC.

Forbes, C., Evans, M. Hastings, N., & Peacock, B. (2011). Statistical Distributions. John Wiley & Sons.

Examples

x <- rrayleigh(1e5, 13)
xx <- seq(-100, 100, by = 0.001)
hist(x, 100, freq = FALSE)
lines(xx, drayleigh(xx, 13), col = "red")
hist(prayleigh(x, 13)) 
plot(ecdf(x))
lines(xx, prayleigh(xx, 13), col = "red", lwd = 2)