drayleigh {bayesmeta} | R Documentation |
Rayleigh density, distribution, quantile function, random number generation, and expectation and variance.
drayleigh(x, scale=1, log=FALSE) prayleigh(q, scale=1) qrayleigh(p, scale=1) rrayleigh(n, scale=1) erayleigh(scale=1) vrayleigh(scale=1)
x, q |
quantile. |
p |
probability. |
n |
number of observations. |
scale |
scale parameter (>0). |
log |
logical; if |
The Rayleigh distribution arises as the distribution of the square root of an exponentially distributed (or χ^2_2-distributed) random variable. If X follows an exponential distribution with rate λ and expectation 1/λ, then Y=sqrt(X) follows a Rayleigh distribution with scale sigma=1/sqrt(2*lambda) and expectation sqrt(pi/(4*lambda)).
Note that the exponential distribution is the maximum entropy distribution among distributions supported on the positive real numbers and with a pre-specified expectation; so the Rayleigh distribution gives the corresponding distribution of its square root.
‘drayleigh()
’ gives the density function,
‘prayleigh()
’ gives the cumulative distribution
function (CDF),
‘qrayleigh()
’ gives the quantile function (inverse CDF),
and ‘rrayleigh()
’ generates random deviates.
The ‘erayleigh()
’ and ‘vrayleigh()
’
functions return the corresponding Rayleigh distribution's
expectation and variance, respectively.
Christian Roever christian.roever@med.uni-goettingen.de
C. Roever, R. Bender, S. Dias, C.H. Schmid, H. Schmidli, S. Sturtz, S. Weber, T. Friede. On weakly informative prior distributions for the heterogeneity parameter in Bayesian random-effects meta-analysis. Research Synthesis Methods, 12(4):448-474, 2021. doi: 10.1002/jrsm.1475.
N.L. Johnson, S. Kotz, N. Balakrishnan. Continuous univariate distributions, volume 1. Wiley, New York, 2nd edition, 1994.
dexp
, dlomax
,
dhalfnormal
, dhalft
, dhalfcauchy
,
TurnerEtAlPrior
, RhodesEtAlPrior
,
bayesmeta
.
######################## # illustrate densities: x <- seq(0,6,le=200) plot(x, drayleigh(x, scale=0.5), type="l", col="green", xlab=expression(tau), ylab=expression("probability density "*f(tau))) lines(x, drayleigh(x, scale=1/sqrt(2)), col="red") lines(x, drayleigh(x, scale=1), col="blue") abline(h=0, v=0, col="grey") ############################################### # illustrate exponential / Rayleigh connection # via a quantile-quantile plot (Q-Q-plot): N <- 10000 exprate <- 5 plot(sort(sqrt(rexp(N, rate=exprate))), qrayleigh(ppoints(N), scale=1/sqrt(2*exprate))) abline(0, 1, col="red") ############################################### # illustrate Maximum Entropy distributions # under similar but different constraints: mu <- 0.5 tau <- seq(0, 4*mu, le=100) plot(tau, dexp(tau, rate=1/mu), type="l", col="red", ylim=c(0,1/mu), xlab=expression(tau), ylab="probability density") lines(tau, drayleigh(tau, scale=1/sqrt(2*1/mu^2)), col="blue") abline(h=0, v=0, col="grey") abline(v=mu, col="darkgrey"); axis(3, at=mu, label=expression(mu)) # explicate constraints: legend("topright", pch=15, col=c("red","blue"), c(expression("Exponential: E["*tau*"]"==mu), expression("Rayleigh: E["*tau^2*"]"==mu^2)))