## Poisson sampling with a gamma prior

### Description

Evaluates and plots the posterior density for mu, the mean rate of occurance in a Poisson process and a gamma prior on mu

### Usage

```poisgamp(y, shape, rate = 1, scale = 1/rate, alpha = 0.05, ...)
```

### Arguments

 `y` a random sample from a Poisson distribution. `shape` the shape parameter of the gamma prior. `rate` the rate parameter of the gamma prior. Note that the scale is 1 / rate `scale` the scale parameter of the gamma prior `alpha` the width of the credible interval is controlled by the parameter alpha. `...` additional arguments that are passed to `Bolstad.control`

### Value

An object of class 'Bolstad' is returned. This is a list with the following components:

 `prior` the prior density assigned to mu `likelihood` the scaled likelihood function for mu given y `posterior` the posterior probability of mu given y `shape` the shape parameter for the gamma posterior `rate` the rate parameter for the gamma posterior

`poisdp` `poisgcp`

### Examples

```
## simplest call with an observation of 4 and a gamma(1, 1), i.e. an exponential prior on the
## mu
poisgamp(4, 1, 1)

##  Same as the previous example but a gamma(10, ) prior
poisgamp(4, 10, 1)

##  Same as the previous example but an improper gamma(1, ) prior
poisgamp(4, 1, 0)

## A random sample of 50 observations from a Poisson distribution with
## parameter mu = 3 and  gamma(6,3) prior
set.seed(123)
y = rpois(50,3)
poisgamp(y,6,3)

## In this example we have a random sample from a Poisson distribution
## with an unknown mean. We will use a gamma(6,3) prior to obtain the
## posterior gamma distribution, and use the R function qgamma to get a
## 95% credible interval for mu
y = c(3,4,4,3,3,4,2,3,1,7)
results = poisgamp(y,6,3)
ci = qgamma(c(0.025,0.975),results\$shape, results\$rate)
cat(paste("95% credible interval for mu: [",round(ci,3), ",", round(ci,3)),"]\n")

## In this example we have a random sample from a Poisson distribution
## with an unknown mean. We will use a gamma(6,3) prior to obtain the
## posterior gamma distribution, and use the R function qgamma to get a
## 95% credible interval for mu
y = c(3,4,4,3,3,4,2,3,1,7)
results = poisgamp(y, 6, 3)
ci = quantile(results, c(0.025, 0.975))
cat(paste("95% credible interval for mu: [",round(ci,3), ",", round(ci,3)),"]\n")

```