Rayleigh function

Rayleigh distribution

Rayleigh distribution

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

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≤x]P[X \le x]

    otherwise, P[X>x]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)=xσ2exp⁡(−x22σ2)f(x)=x/σ2∗exp(−(x2/2∗σ2)) f(x) = \frac{x}{\sigma^2} \exp\left(-\frac{x^2}{2\sigma^2}\right)f(x) = x/\sigma^2 * exp(-(x^2 / 2*\sigma^2))

Cumulative distribution function

F(x)=1−exp⁡(−x22σ2)F(x)=1−exp(−x2/2∗σ2) F(x) = 1 - \exp\left(-\frac{x^2}{2\sigma^2}\right)F(x) = 1 - exp(-x^2 / 2*\sigma^2)

Quantile function

F−1(p)=−2σ2log⁡(1−p)F−1(p)=sqrt(−2∗σ2∗log(1−p)) F^{-1}(p) = \sqrt{-2\sigma^2 \log(1-p)}F^-1(p) = sqrt(-2*\sigma^2 * log(1-p))

Examples

x <- rrayleigh(1e5, 13)
hist(x, 100, freq = FALSE)
curve(drayleigh(x, 13), 0, 60, col = "red", add = TRUE)
hist(prayleigh(x, 13)) 
plot(ecdf(x))
curve(prayleigh(x, 13), 0, 60, col = "red", lwd = 2, add = TRUE)

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.

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  • Maintainer: Tymoteusz Wolodzko
  • License: GPL-2
  • Last published: 2023-11-30

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