Rayleigh {shotGroups}

The Rayleigh Distribution

Description

Density, distribution function, quantile function, and random deviate generation for the Rayleigh distribution. The radius around the true mean in a bivariate uncorrelated normal random variable with equal variances, re-written in polar coordinates (radius and angle), follows a Rayleigh distribution.

Usage

dRayleigh(x, scale)
pRayleigh(q, scale, lower.tail = TRUE)
qRayleigh(p, scale, lower.tail = TRUE)
rRayleigh(n, scale)

Arguments

x, q

vector of quantiles.

p

vector of probabilities.

n

number of observations. If length(n) > 1, the length is taken to be the number required.

scale

vector of Rayleigh scale parameters (common standard deviation of the underlying normal distribution of each 2D-coordinate).

lower.tail

logical. If TRUE (default), probabilities are P[X \le x] otherwise, P[X > x].

Details

The parameter scale may be determined with getRayParam.

See Maxwell for the distribution of radial error around the true center of uncorrelated trivariate normal variables with equal variances. See Hoyt for the distribution of radial error around the true center of correlated bivariate normal variables with unequal variances. See Rice for the distribution of radial error around an offset center for uncorrelated bivariate normal variables with equal variances. See mvnEll for the distribution of radial error around an offset center for correlated normal variables with unequal variances.

Value

dRayleigh gives the density, pRayleigh gives the cumulative distribution function, qRayleigh gives the quantile function, 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 parameters for the other functions.

In dRayleigh, pRayleigh and qRayleigh, the numerical parameters are recycled to the length of the result. Only the first element of the logical parameters is used. In rRayleigh, only the first element of scale is used.

References

https://reference.wolfram.com/language/ref/RayleighDistribution.html

See Also

getRayParam, Maxwell, Rice, Hoyt, mvnEll

Examples

dRayleigh(1, scale=10)
pRayleigh(c(0.1, 0.5, 0.9), scale=10)
qRayleigh(0.5, scale=c(5, 10, 15))
rRayleigh(5, scale=10)