drayleigh {bayesmeta} | R Documentation |
The Rayleigh distribution.
Description
Rayleigh density, distribution, quantile function, random number generation, and expectation and variance.
Usage
drayleigh(x, scale=1, log=FALSE)
prayleigh(q, scale=1)
qrayleigh(p, scale=1)
rrayleigh(n, scale=1)
erayleigh(scale=1)
vrayleigh(scale=1)
Arguments
x , q |
quantile. |
p |
probability. |
n |
number of observations. |
scale |
scale parameter ( |
log |
logical; if |
Details
The Rayleigh distribution arises as the distribution of the
square root of an exponentially distributed (or
\chi^2_2
-distributed) random variable.
If X
follows an exponential distribution with rate \lambda
and expectation 1/\lambda
, 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.
Value
‘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.
Author(s)
Christian Roever christian.roever@med.uni-goettingen.de
References
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.
See Also
dexp
, dlomax
,
dhalfnormal
, dhalft
, dhalfcauchy
,
TurnerEtAlPrior
, RhodesEtAlPrior
,
bayesmeta
.
Examples
########################
# 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)))