dist.Inverse.Gamma {LaplacesDemon} | R Documentation |

## Inverse Gamma Distribution

### Description

This is the density function and random generation from the inverse gamma distribution.

### Usage

```
dinvgamma(x, shape=1, scale=1, log=FALSE)
rinvgamma(n, shape=1, scale=1)
```

### Arguments

`n` |
This is the number of draws from the distribution. |

`x` |
This is the scalar location to evaluate density. |

`shape` |
This is the scalar shape parameter |

`scale` |
This is the scalar scale parameter |

`log` |
Logical. If |

### Details

Application: Continuous Univariate

Density:

`p(\theta) = \frac{\beta^\alpha}{\Gamma(\alpha)} \theta^{-(\alpha + 1)} \exp(-\frac{\beta}{\theta}), \quad \theta > 0`

Inventor: Unknown (to me, anyway)

Notation 1:

`\theta \sim \mathcal{G}^{-1}(\alpha, \beta)`

Notation 2:

`p(\theta) = \mathcal{G}^{-1}(\theta | \alpha, \beta)`

Parameter 1: shape

`\alpha > 0`

Parameter 2: scale

`\beta > 0`

Mean:

`E(\theta) = \frac{\beta}{\alpha - 1}`

, for`\alpha > 1`

Variance:

`var(\theta) = \frac{\beta^2}{(\alpha - 1)^2 (\alpha - 2)}, \alpha > 2`

Mode:

`mode(\theta) = \frac{\beta}{\alpha + 1}`

The inverse-gamma is the conjugate prior distribution for the normal
or Gaussian variance, and has been traditionally specified as a vague
prior in that application. The density is always finite; its integral is
finite if `\alpha > 0`

. Prior information decreases as
`\alpha, \beta \rightarrow 0`

.

These functions are similar to those in the `MCMCpack`

package.

### Value

`dinvgamma`

gives the density and
`rinvgamma`

generates random deviates. The parameterization
is consistent with the Gamma Distribution in the stats package.

### See Also

`dgamma`

,
`dnorm`

,
`dnormp`

, and
`dnormv`

.

### Examples

```
library(LaplacesDemon)
x <- dinvgamma(4.3, 1.1)
x <- rinvgamma(10, 3.3)
#Plot Probability Functions
x <- seq(from=0.1, to=20, by=0.1)
plot(x, dinvgamma(x,1,1), ylim=c(0,1), type="l", main="Probability Function",
ylab="density", col="red")
lines(x, dinvgamma(x,1,0.6), type="l", col="green")
lines(x, dinvgamma(x,0.6,1), type="l", col="blue")
legend(2, 0.9, expression(paste(alpha==1, ", ", beta==1),
paste(alpha==1, ", ", beta==0.6), paste(alpha==0.6, ", ", beta==1)),
lty=c(1,1,1), col=c("red","green","blue"))
```

*LaplacesDemon*version 16.1.6 Index]