Density, distribution function, quantile function and random generation
for the Rayleigh distribution with scale parameter sigma.
Usage
drayleigh(x, sigma, log = FALSE)
prayleigh(q, sigma, lower.tail = TRUE, log.p = FALSE)
qrayleigh(p, sigma, lower.tail = TRUE, log.p = FALSE)
rrayleigh(n, sigma)Arguments
- x, q
vector of quantiles.
- sigma
scale parameter. Must be positive.
- 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
drayleigh gives the density, prayleigh gives the distribution function,
qrayleigh gives the quantile function, and rrayleigh generates random
deviates.
The length of the result is determined by n for rrayleigh, and is the
maximum of the lengths of the numerical arguments for the other functions.
Details
The Rayleigh distribution is a continuous probability distribution for non-negative random variables. It arises as the distribution of the magnitude of a two-dimensional vector whose components are independent, identically distributed Gaussian random variables with zero mean.
The probability density function is given by: $$f(x) = \frac{x}{\sigma^2} \exp\left(-\frac{x^2}{2\sigma^2}\right)$$ for \(x \geq 0\) and \(\sigma > 0\).
The cumulative distribution function is: $$F(x) = 1 - \exp\left(-\frac{x^2}{2\sigma^2}\right)$$
References
Rayleigh, Lord (1880). On the resultant of a large number of vibrations of the same pitch and of arbitrary phase. Philosophical Magazine, 10(60), 73-78.
Examples
# Density at x = 1 with sigma = 1
drayleigh(1, sigma = 1)
#> [1] 0.6065307
# CDF at x = 1
prayleigh(1, sigma = 1)
#> [1] 0.3934693
# Quantile for p = 0.5 (median)
qrayleigh(0.5, sigma = 1)
#> [1] 1.17741
# Generate 10 random values
rrayleigh(10, sigma = 1)
#> [1] 0.8154869 0.9253405 1.6880767 2.2109872 1.5589187 1.1066728 0.7037479
#> [8] 1.6140235 1.5154609 0.3279180