Normal/Gaussian Probability Distribution Family

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):
  • a Binomial Probability Distribution Family.
  • a Poisson Probability Distribution Family.
• See: Binomial Statistical Model, Multinomial Distribution.

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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.

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