Alternative Names
- normal distribution
- Gaussian distribution
- Error distribution
Interactive Distribution Calculator
Type
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
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
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
Variance
Skewness
Excess Kurtosis
Entropy
- \(\frac{1}{2} \ln(2 \pi e \sigma^2)\)
Median
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