Alternative Names

• normal distribution
• Gaussian distribution
• Error distribution

Interactive Distribution Calculator

• Interactive Java Distribution Applet
  

Type

• Continuous

Model

• The normal distribution is used to model physical quantities that are subject to numerous small, random errors.

Parameters

• \(\mu \in (-\infty, \infty)\), the location parameter
• \(\sigma \in (0, \infty)\), the scale parameter

Support

• \((-\infty, \infty)\)

Density function

• \(f(x)=\frac{1}{\sqrt{2 \pi} \sigma} \exp \left(-\frac{1}{2}(\frac{x - \mu}{\sigma})^2 \right), \; x \in (-\infty, \infty)\)

Mode

• \(\mu\)

Distribution Function

• \( F(x) = \Phi\left(\frac{x - \mu}{\sigma}\right), \; x \in (-\infty, \infty) \) where \(\Phi\) is the standard normal distribution function

Quantile Function

• \( Q(p) = \mu + \sigma \Phi^{-1}(p), \; p \in (0, 1) \) where \(\Phi\) is the standard normal distribution function

Moment Generating Function

• \(M(t) = \exp(\mu t + \frac{1}{2} \sigma^2 t^2), \; t \in (-\infty, \infty)\)

Characteristic Function

• \(\varphi(t) = \exp(i \mu t - \frac{1}{2} \sigma^2 t^2), \; t \in (-\infty, \infty)\)

Mean

• \(\mu\)

Variance

• \(\sigma^2\)

Skewness

• 0

Excess Kurtosis

• 0

Entropy

• \(\frac{1}{2} \ln(2 \pi e \sigma^2)\)

Median

• \(\mu\)

First Quartile

• \(\mu - \Phi^{-1}(\frac{1}{4}) \sigma\) where \(\Phi\) is the standard normal distribution function

Third Quartile

• \(\mu - \Phi^{-1}(\frac{3}{4}) \sigma\) where \(\Phi\) is the standard normal distribution function

Families

• location
• scale
• general exponential
• stable