# Normal/Gaussian Probability Distribution Family

(Redirected from normal distribution)

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

**AKA:**[math]\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]\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