# Normal/Gaussian Probability Distribution Family

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A Normal/Gaussian Probability Distribution Family is an exponential probability distribution family whose exponential function is of the form [math]\displaystyle{ f(x,a,b,c) }[/math] (where [math]\displaystyle{ a = \tfrac{1}{\sqrt{2\pi\sigma^2}} }[/math], [math]\displaystyle{ b = \mu }[/math], and [math]\displaystyle{ c = 2\sigma^2 }[/math].

**AKA:**[math]\displaystyle{ \mathcal{N}(x | \mu, \sigma) }[/math]**Context:**- It can be instantiated as a Gaussian Probability Density Function (or a Gaussian dataset), such as a standard Gaussian [math]\displaystyle{ \mathcal{N}(x | \mu=0, \sigma=1) }[/math].
- It can range from being a Univariate Gaussian Distribution to being a Multivariate Gaussian Distribution.
- It can be a member of a Gaussian Mixture Model.
- …

**Counter-Example(s):****See:**Binomial Statistical Model, Multinomial Distribution.

## References

### 2013

- http://en.wikipedia.org/wiki/Normal_distribution#Operations_on_normal_deviates
- The family of normal distributions is closed under linear transformations. That is, if
*X*is normally distributed with mean μ*and deviation*σ*, then the variable*Y*=*aX +*b*, for any real numbers*a*and*b*, is also normally distributed, with mean*aμ*+*b*and deviation*aσ*.

- The family of normal distributions is closed under linear transformations. That is, if