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Normal-gamma distribution: Information from Answers.com

Normal-gamma
Probability density function
Cumulative distribution function
Parameters \mu\, location (real)
\lambda > 0\, (real)
\alpha \ge 1\, (real)
\beta \ge 0\, (real)
Support x \in (-\infty, \infty)\,\!, \; \tau \in (0,\infty)
Probability density function (pdf)
Cumulative distribution function (cdf)
Mean \mu\,\!
Median \mu\,
Mode
Variance \left(\frac{\lambda+1}{\lambda}\right)\frac{\beta}{\alpha-1}\,\!
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 precision.

Definition

Suppose

x|\tau, \mu, \lambda \sim N(\mu,\lambda / \tau) \,\!

has a normal distribution with mean μ and variance λ / τ, where

\tau|\alpha, \beta \sim \mathrm{Gamma}(\alpha,\beta) \!

has a gamma distribution. Then (x,τ) has a normal-gamma distribution, denoted as

(x,\tau) \sim \mathrm{NormalGamma}(\mu,\lambda,\alpha,\beta) \! .

Characterization

Probability density function

f(x,\tau|\mu,\lambda,\alpha,\beta) = \frac{\beta^\alpha}{\Gamma(\alpha)\sqrt{2\pi\lambda}} \, \tau^{\alpha-\frac{1}{2}}\,e^{-\beta\tau}\,e^{ -\frac{\tau(x- \mu)^2}{2\lambda}}

Cumulative distribution function

Properties

Summation

Scaling

For any t > 0, tX is distributed NormalGamma(tμ,λ,α,t2β)

Exponential family

Information entropy

Kullback-Leibler divergence

Maximum likelihood estimation

Generating normal-gamma random variates

Generation of random variates is straightforward:

  1. Sample τ from a gamma distribution with parameters α and β
  2. Sample x from a normal distribution with mean μ and variance λ / τ

Related distributions

References

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

v • d • e

Probability distributions
Discrete univariate with finite support

Benford · Bernoulli · binomial · categorical · hypergeometric · Rademacher · discrete uniform · Zipf · Zipf-Mandelbrot
Discrete univariate with infinite support
Continuous univariate supported on a bounded interval, e.g. [0,1]
Continuous univariate supported on a semi-infinite interval, usually [0,∞)
Continuous univariate supported on the whole real line (-∞,∞)
Multivariate (joint)
Directional, degenerate, and singular
Families

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