lmomco (version 2.4.14)

pdftexp: Probability Density Function of the Truncated Exponential Distribution

Description

This function computes the probability density of the Truncated Exponential distribution given parameters (\(\psi\) and \(\alpha\)) computed by partexp. The parameter \(\psi\) is the right truncation, and \(\alpha\) is a scale parameter. The probability density function, letting \(\beta = 1/\alpha\) to match nomenclature of Vogel and others (2008), is $$f(x) = \frac{\beta\,\exp(-\beta{t})}{1 - \mathrm{exp}(-\beta\psi)}\mbox{,}$$ where \(x(x)\) is the probability density for the quantile \(0 \le x \le \psi\) and \(\psi > 0\) and \(\alpha > 0\). This distribution represents a nonstationary Poisson process.

The distribution is restricted to a narrow range of L-CV (\(\tau_2 = \lambda_2/\lambda_1\)). If \(\tau_2 = 1/3\), the process represented is a stationary Poisson for which the probability density function is simply the uniform distribution and \(f(x) = 1/\psi\). If \(\tau_2 = 1/2\), then the distribution is represented as the usual exponential distribution with a location parameter of zero and a scale parameter \(1/\beta\). Both of these limiting conditions are supported.

Usage

pdftexp(x, para)

Value

Probability density (\(F\)) for \(x\).

Arguments

x

A real value vector.

para

The parameters from partexp or vec2par.

Author

W.H. Asquith

References

Vogel, R.M., Hosking, J.R.M., Elphick, C.S., Roberts, D.L., and Reed, J.M., 2008, Goodness of fit of probability distributions for sightings as species approach extinction: Bulletin of Mathematical Biology, DOI 10.1007/s11538-008-9377-3, 19 p.

See Also

cdftexp, quatexp, lmomtexp, partexp

Examples

lmr <- vec2lmom(c(40,0.38), lscale=FALSE)
pdftexp(0.5,partexp(lmr))
if (FALSE) {
F <- seq(0,1,by=0.001)
A <- partexp(vec2lmom(c(100, 1/2), lscale=FALSE))
x <- quatexp(F, A)
plot(x, pdftexp(x, A), pch=16, type='l')
by <- 0.01; lcvs <- c(1/3, seq(1/3+by, 1/2-by, by=by), 1/2)
reds <- (lcvs - 1/3)/max(lcvs - 1/3)
for(lcv in lcvs) {
    A <- partexp(vec2lmom(c(100, lcv), lscale=FALSE))
    x <- quatexp(F, A)
    lines(x, pdftexp(x, A),
          pch=16, col=rgb(reds[lcvs == lcv],0,0))
}
}