Inverse-chi-squared distribution
Probability distribution From Wikipedia, the free encyclopedia
In probability and statistics, the inverse-chi-squared distribution (or inverted-chi-square distribution[1]) is a continuous probability distribution of a positive-valued random variable. It is closely related to the chi-squared distribution. It is used in Bayesian inference as conjugate prior for the variance of the normal distribution.[2]
Probability density function ![]() | |||
Cumulative distribution function ![]() | |||
Parameters | |||
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Support | |||
CDF | |||
Mean | for | ||
Median | |||
Mode | |||
Variance | for | ||
Skewness | for | ||
Excess kurtosis | for | ||
Entropy |
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MGF | ; does not exist as real valued function | ||
CF |
Definition
Summarize
Perspective
The inverse chi-squared distribution (or inverted-chi-square distribution[1] ) is the probability distribution of a random variable whose multiplicative inverse (reciprocal) has a chi-squared distribution.
If follows a chi-squared distribution with degrees of freedom then follows the inverse chi-squared distribution with degrees of freedom.
The probability density function of the inverse chi-squared distribution is given by
In the above and is the degrees of freedom parameter. Further, is the gamma function.
The inverse chi-squared distribution is a special case of the inverse-gamma distribution. with shape parameter and scale parameter .
Related distributions
- chi-squared: If and , then
- scaled-inverse chi-squared: If , then
- Inverse gamma with and
- Inverse chi-squared distribution is a special case of type 5 Pearson distribution
See also
References
External links
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