Inverse-chi-squared distribution

Inverse-chi-squared distribution is a probability distribution that is closely related to the chi-squared distribution. It is a particular case of the inverse gamma distribution. The inverse-chi-squared distribution is used extensively in Bayesian statistics, statistical inference, and hypothesis testing. It is especially important in the context of estimating the variance of a normally distributed population when the mean is known.

Definition

The inverse-chi-squared distribution with degrees of freedom \( \nu \) is defined as the distribution of \( \frac{1}{X} \) where \( X \) follows a chi-squared distribution with \( \nu \) degrees of freedom. Mathematically, if \( X \sim \chi^2(\nu) \), then \( \frac{1}{X} \) follows an inverse-chi-squared distribution, denoted as \( X \sim \text{Inv-}\chi^2(\nu) \).

Probability Density Function

The probability density function (pdf) of an inverse-chi-squared distribution with \( \nu \) degrees of freedom is given by:

\[ f(x; \nu) = \frac{2^{-\nu/2}}{\Gamma(\nu/2)} x^{-\nu/2-1} e^{-1/(2x)} \]

for \( x > 0 \), where \( \Gamma \) is the Gamma function.

Properties

Mean

The mean of the inverse-chi-squared distribution is \( \frac{1}{\nu-2} \) for \( \nu > 2 \).

Variance

The variance is \( \frac{2}{(\nu-2)^2(\nu-4)} \) for \( \nu > 4 \).

Applications

The inverse-chi-squared distribution is used in Bayesian statistics as the conjugate prior for the variance parameter of a normal distribution. This means that if the prior distribution of the variance is an inverse-chi-squared distribution, then the posterior distribution, after observing data, will also be an inverse-chi-squared distribution. This property simplifies the process of updating beliefs about the variance in light of new data.

Related Distributions

• The chi-squared distribution is the distribution of \( X \) when \( \frac{1}{X} \) follows an inverse-chi-squared distribution.
• The inverse gamma distribution is a generalization of the inverse-chi-squared distribution.
• The scaled-inverse-chi-squared distribution is another distribution related to the inverse-chi-squared distribution, often used in Bayesian statistics for similar purposes.

See Also

• Bayesian statistics
• Statistical inference
• Gamma function
• Conjugate prior distributions