Definition:Gaussian Distribution
(Redirected from Definition:Normal Distribution)
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Definition
Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.
Then $X$ has a Gaussian distribution if and only if the probability density function of $X$ is:
- $\map {f_X} x = \dfrac 1 {\sigma \sqrt {2 \pi} } \map \exp {-\dfrac {\paren {x - \mu}^2} {2 \sigma^2} }$
for $\mu \in \R, \sigma \in \R_{> 0}$.
This is written:
- $X \sim \Gaussian \mu {\sigma^2}$
Also known as
The Gaussian distribution is also commonly known as the normal distribution (hence the notation).
The former term is preferred on $\mathsf{Pr} \infty \mathsf{fWiki}$.
Also see
- Expectation of Gaussian Distribution: $\expect X = \mu$
- Variance of Gaussian Distribution: $\var X = \sigma^2$
- Results about the Gaussian distribution can be found here.
Source of Name
This entry was named for Carl Friedrich Gauss.
Technical Note
The $\LaTeX$ code for \(\Gaussian {\mu} {\sigma^2}\) is \Gaussian {\mu} {\sigma^2}
.
When either argument is a single character, it is usual to omit the braces:
\Gaussian \mu {\sigma^2}
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Gaussian curvature
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.25$: Gauss ($\text {1777}$ – $\text {1855}$)
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Gaussian distribution
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Gaussian distribution
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Gaussian distribution
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $13$: Probability distributions
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $15$: Probability distributions
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- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 39$: Probability Distributions: Normal Distribution: $39.4$