# Normal-gamma distribution

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Parameters Probability density function Cumulative distribution function ${\displaystyle \mu \,}$ location (real)${\displaystyle \lambda >0\,}$ (real)${\displaystyle \alpha \geq 1\,}$ (real)${\displaystyle \beta \geq 0\,}$ (real) ${\displaystyle x\in (-\infty ,\infty )\,\!,\;\tau \in (0,\infty )}$ ${\displaystyle \mu \,\!}$ ${\displaystyle \mu \,}$ ${\displaystyle \left({\frac {\lambda +1}{\lambda }}\right){\frac {\beta }{\alpha -1}}\,\!}$

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

${\displaystyle x|\tau ,\mu ,\lambda \sim N(\mu ,\lambda /\tau )\,\!}$

has a normal distribution with mean ${\displaystyle \mu }$ and variance ${\displaystyle \lambda /\tau }$, where

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

has a gamma distribution. Then ${\displaystyle (x,\tau )}$ has a normal-gamma distribution, denoted as

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

## Characterization

### Probability density function

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

## Properties

### Scaling

For any t > 0, tX is distributed ${\displaystyle {\rm {NormalGamma}}(t\mu ,\lambda ,\alpha ,t^{2}\beta )}$

## References

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