sdprisk (version 1.0-3)

hypoexp: Hypo-Exponential Distribution

Description

Density, distribution function, random generation and moment-generating function (and its first two derivatives) for the hypo-exponential distribution with rates rate.

Usage

dhypoexp(x, rate = 1, log = FALSE)
phypoexp(q, rate = 1, lower.tail = TRUE, log.p = FALSE, tailarea = FALSE)
rhypoexp(n = 1, rate = 1)
mgfhypoexp(x, rate = 1, difforder = 0)

Arguments

x, q

vector of quantiles.

n

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

difforder

the order of derivative for the moment-generating function; currently only implemented for 0, 1, 2.

rate

vector of (unique) rates.

lower.tail

logical; if TRUE, probabilities are $\mathbf{P}(X \le x)$, otherwise $\mathbf{P}(X > x)$.

log, log.p

logical; if TRUE, probabilities $p$ are given as $\log(p)$.

tailarea

logical; if TRUE, probabilities are given for the integrated tail area distribution.

Value

• dhypoexp gives the density, phypoexp gives the distribution function (or the integrated tail area distribution function), rhypoexp generates random deviates and mgfhypoexp gives the moment-generating function (or its derivative up to the second order)

Details

The sum of $n$ independent exponentially distributed random variables $X_{i}$ with rate parameters $\lambda_{i}$ has a hypo-exponential distribution with rate vector $(\lambda_{1}, \dots, \lambda_{n})$.

The hypo-exponential distribution is a generalization of the Erlang distribution (a Gamma distribution with an integer-valued shape parameter) and a special case of the phase-type distribution (see References section).

References

Neuts, M. F. (1981) Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach, reprinted and corrected.

See Also

dexp, dgamma

Examples

## Random generation
rhypoexp(10, c(3, 5))

## Mean
mu <- mgfhypoexp(0, c(3, 5), difforder = 1)

## Variance
mgfhypoexp(0, c(3, 5), difforder = 2) - mu^2