Definition:Normal Distribution

Definition

Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.

Then $X$ has a normal 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 normal distribution is also commonly known as the Gaussian distribution, for Carl Friedrich Gauss.

Contrary to the usual house style, the former term is preferred on $\mathsf{Pr} \infty \mathsf{fWiki}$.

A popular term for the normal distribution is bell-shaped, from the shape of the graph of its frequency function.

Also see

• Expectation of Normal Distribution: $\expect X = \mu$
• Variance of Normal Distribution: $\var X = \sigma^2$

• Definition:Standard Normal Distribution

• Results about the normal distribution can be found here.

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

• 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): distribution
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): normal distribution
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): distribution
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): normal 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: $39.4$