Normal-gamma distribution

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Normal-gamma
Probability density function
Cumulative distribution function
Parameters location (real)
(real)
(real)
(real)
Support
Probability density function (pdf)
Cumulative distribution function (cdf)
Mean
Median
Mode
Variance
Skewness
Excess kurtosis
Entropy
Moment-generating function (mgf)
Characteristic function

In probability theory and statistics, the normal-gamma distribution is a four-parameter family of continuous probability distributions. It is the conjugate prior of a normal distribution with unknown mean and variance.

Definition

Suppose

has a normal distribution with mean and variance , where

has a gamma distribution. Then has a normal-gamma distribution, denoted as

Characterization

Probability density function

Cumulative distribution function

Properties

Summation

Scaling

For any t > 0, tX is distributed

Exponential family

Information entropy

Kullback-Leibler divergence

Maximum likelihood estimation

Generating normal-gamma random variates

References

  • Bernardo, J. M., and A. F. M. Smith. 1994. Bayesian theory. Chichester, UK: Wiley.
  • Dearden et al. Bayesian Q-learning

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