## The Gamma Distribution

### Description

Density, distribution function, quantile function and random generation for the Gamma distribution with parameters `shape` and `scale`.

### Usage

```dgamma(x, shape, rate = 1, scale = 1/rate, log = FALSE)
pgamma(q, shape, rate = 1, scale = 1/rate, lower.tail = TRUE,
log.p = FALSE)
qgamma(p, shape, rate = 1, scale = 1/rate, lower.tail = TRUE,
log.p = FALSE)
rgamma(n, shape, rate = 1, scale = 1/rate)
```

### Arguments

 `x, q` vector of quantiles. `p` vector of probabilities. `n` number of observations. If `length(n) > 1`, the length is taken to be the number required. `rate` an alternative way to specify the scale. `shape, scale` shape and scale parameters. Must be strictly positive. `log, log.p` logical; if `TRUE`, probabilities/densities p are returned as log(p). `lower.tail` logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].

### Details

If `scale` is omitted, it assumes the default value of `1`.

The Gamma distribution with parameters `shape` = a and `scale` = s has density

f(x)= 1/(s^a Gamma(a)) x^(a-1) e^-(x/s)

for x > 0, a > 0 and s > 0. (Here Gamma(a) is the function implemented by R's `gamma()` and defined in its help.)

The mean and variance are E(X) = a*s and Var(X) = a*s^2.

The cumulative hazard H(t) = - log(1 - F(t)) is `-pgamma(t, ..., lower = FALSE, log = TRUE)`.

### Value

`dgamma` gives the density, `pgamma` gives the distribution function, `qgamma` gives the quantile function, and `rgamma` generates random deviates.

### Note

The S parametrization is via `shape` and `rate`: S has no `scale` parameter.

`pgamma` is closely related to the incomplete gamma function. As defined by Abramowitz and Stegun 6.5.1

P(a,x) = 1/Gamma(a) integral_0^x t^(a-1) exp(-t) dt

P(a, x) is `pgamma(x, a)`. Other authors (for example Karl Pearson in his 1922 tables) omit the normalizing factor, defining the incomplete gamma function as `pgamma(x, a) * gamma(a)`.

As from R 2.1.0 `pgamma()` uses a new algorithm (mainly by Morten Welinder) which should be uniformly as accurate as AS 239.

### References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.

Shea, B. L. (1988) Algorithm AS 239, Chi-squared and Incomplete Gamma Integral, Applied Statistics (JRSS C) 37, 466–473.

Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions. New York: Dover. Chapter 6: Gamma and Related Functions.

`gamma` for the gamma function, `dbeta` for the Beta distribution and `dchisq` for the chi-squared distribution which is a special case of the Gamma distribution.
```-log(dgamma(1:4, shape=1))