## Compound Poisson-Exponential distribution

### Description

Density, distribution function, quantile function and random generation for the Poisson-exponential compound probability distribution with parameters `fraction` and `rate`.

### Usage

```dpoix(x, frac, rate, log=FALSE)
ppoix(q, frac, rate, lower.tail=TRUE, log.p=FALSE)
qpoix(p, frac, rate, lower.tail=TRUE, log.p=FALSE)
rpoix(n, frac, rate)
```

### Arguments

 `x` vector of (non-negative integer) quantiles. In the context of species abundance distributions, this is a vector of abundances of species in a sample. `q` vector of (non-negative integer) quantiles. In the context of species abundance distributions, a vector of abundances of species in a sample. `n` number of random values to return. `p` vector of probabilities. `frac` single numeric 0 < frac <= 1; fraction of the population or community sampled (see details). `rate` vector of (non-negative) rates of the exponential distribution of the sampled population (see details). `log, log.p` logical; if TRUE, probabilities p are given as log(p) `lower.tail` logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].

### Details

A compound Poisson-exponential distribution is a Poisson probability distribution where its single parameter lambda, is frac*n, at which n is a random variable with exponential distribution. Thus, the expected value and variance are E[X] = Var[X] = frac*n . The density function is

p(y) = rate*frac^y / (frac + rate)^(y+1)

for x = 0, 1, 2, ... (Green & Plotkin 2007) In ecology, this distribution gives the probability that a species has an abundance of x individuals in a random sample of a fraction `frac` of the community. In the community the species abundances are independent random variables that follow an exponential density function.

Hence, a Poisson-exponential distribution is a model for species abundances distributions (SAD) in a sample taken from a community under the assumptions: (a) species abundances in the community are independent identically distributed exponential variables, (b) sampling is a Poisson process with expected value 'frac*n', (c) individuals are sampled with replacement, or the fraction of total individuals sampled is small enough to approximate a sample with replacement. See Engen (1977) and Alonso et al. (2008) for critic evaluations.

Notice that the Poisson-exponential can be seen as a different form for the MacArthur's Broken stick model (Baczkowski, 2000), so instead of fitting to a Poisson-exponential distribution directly, the user should use `fitbs`.

### Value

(log) density of the (zero-truncated) density.

### References

Alonso, D. and Ostling, A., and Etienne, R.S. 2008. The implicit assumption of symmetry and the species abundance distribution. Ecology Letters, 11: 93–105.

Engen, S. 1977. Comments on two different approaches to the analysis of species frequency data. Biometrics, 33: 205–213.

Pielou, E.C. 1977. Mathematical Ecology. New York: John Wiley and Sons.

Green,J. and Plotkin, J.B. 2007 A statistical theory for sampling species abundances. Ecology Letters 10:1037–1045