Maxwell {shotGroups}

The Maxwell-Boltzmann Distribution

Description

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

Usage

dMaxwell(x, sigma)
pMaxwell(q, sigma, lower.tail = TRUE)
qMaxwell(p, sigma, lower.tail = TRUE)
rMaxwell(n, sigma)

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.

sigma

vector of parameter sigma (common standard deviation of the underlying normal distribution of each 3D-coordinate).

lower.tail

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

Details

The parameter sigma may be determined with getRayParam.

See Rayleigh for the distribution of radial error around the true center of uncorrelated bivariate 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

dMaxwell gives the density, pMaxwell gives the cumulative distribution function, qMaxwell gives the quantile function, rMaxwell generates random deviates.

The length of the result is determined by n for rMaxwell, and is the maximum of the lengths of the numerical parameters for the other functions.

In dMaxwell, pMaxwell and qMaxwell 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 sigma is used.

References

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

See Also

getRayParam, Rayleigh, Hoyt, Rice, mvnEll

Examples

dMaxwell(1, sigma=10)
pMaxwell(c(0.1, 0.5, 0.9), sigma=10)
qMaxwell(0.5, sigma=c(5, 10, 15))
rMaxwell(5, sigma=10)