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# invgamma()

Usage:
 ```invgamma(P, alpha [,upper:T or lower:F]), P and alpha REAL, elements of P between 0 and 1, those of alpha > 0 ```

Keywords: probabilities, random numbers
```invgamma(p,alpha) computes the pth quantile (100*p percent point) of the
gamma distribution with shape parameter alpha.  Its principal use is to
compute critical values for test statistics with a gamma distribution
but may also be used to compute exact confidence intervals for a Poisson
mean.

The elements of p must be between 0 and 1; the elements of alpha must be
positive but need not be integers.

If neither p nor alpha is a scalar (single number), they must be the
same size and shape.  If just one argument is a scalar, it is used to
compute all the elements of the result.

invgamma(p,alpha,upper:T) and invgamma(p,alpha,lower:F) compute the pth
upper tail quantile.  The result is mathematically equivalent to
invgamma(1 - p, alpha) but may be more accurate for small p.

invgamma() is the inverse of cumgamma().

2*invgamma(p,df/2 [,upper:T]) is equivalent to invchi(p,df [,upper:T]).

mu*invgamma(runi(n),alpha)/alpha will generate a random sample of size n
from a gamma distribution with mean mu and shape parameter alpha.

You can use invgamma() to compute an "exact" confidence interval for mu
based on an observed value x_obs of a Poisson random variable with mean
mu.
Cmd> x_obs <- 11 # observed value of x

Cmd> mu_l <- invgamma(.025,x_obs) # lower 95% limit

Cmd> mu_u <- invgamma(.025,x_obs+1,upper:T) # upper 95% limit

Cmd> vector(mu_l,mu_u) # "exact" 95% confidence interval for mu
(1)      5.4912      19.682

Cmd> vector(cumpoi(x_obs,mu_u),cumpoi(x_obs,mu_l,upper:T)) # check
(1)       0.025       0.025