# dist.Inverse.ChiSquare: (Scaled) Inverse Chi-Squared Distribution In LaplacesDemonR/LaplacesDemon: Complete Environment for Bayesian Inference

## Description

This is the density function and random generation for the (scaled) inverse chi-squared distribution.

## Usage

 ```1 2``` ```dinvchisq(x, df, scale, log=FALSE) rinvchisq(n, df, scale=1/df) ```

## Arguments

 `x` This is a vector of quantiles. `n` This is the number of observations. If `length(n) > 1`, then the length is taken to be the number required. `df` This is the degrees of freedom parameter, usually represented as nu. `scale` This is the scale parameter, usually represented as lambda. `log` Logical. If `log=TRUE`, then the logarithm of the density is returned.

## Details

• Application: Continuous Univariate

• Density:

p(theta) = ((nu/2)^(nu/2))/(Γ(nu/2)) lambda^nu (1/theta)^((nu/2)+1) exp(-(nu lambda^2)/(2*theta)), theta >= 0

• Inventor: Derived from the chi-squared distribution

• Notation 1: theta ~ chi^(-2)(nu, lambda)

• Notation 2: p(theta) = chi^(-2)(theta | nu, lambda)

• Parameter 1: degrees of freedom parameter nu > 0

• Parameter 2: scale parameter lambda

• Mean: E(theta) = unknown

• Variance: var(theta) = unknown

• Mode: mode(theta) =

The inverse chi-squared distribution, also called the inverted chi-square distribution, is the multiplicate inverse of the chi-squared distribution. If x has the chi-squared distribution with nu degrees of freedom, then 1 / x has the inverse chi-squared distribution with nu degrees of freedom, and nu / x has the inverse chi-squared distribution with nu degrees of freedom.

These functions are similar to those in the GeoR package.

## Value

`dinvchisq` gives the density and `rinvchisq` generates random deviates.

`dchisq`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13``` ```library(LaplacesDemon) x <- dinvchisq(1,1,1) x <- rinvchisq(10,1) #Plot Probability Functions x <- seq(from=0.1, to=5, by=0.01) plot(x, dinvchisq(x,0.5,1), ylim=c(0,1), type="l", main="Probability Function", ylab="density", col="red") lines(x, dinvchisq(x,1,1), type="l", col="green") lines(x, dinvchisq(x,5,1), type="l", col="blue") legend(3, 0.9, expression(paste(nu==0.5, ", ", lambda==1), paste(nu==1, ", ", lambda==1), paste(nu==5, ", ", lambda==1)), lty=c(1,1,1), col=c("red","green","blue")) ```