Truncated Normal Distribution#

Univariate, Continuous, Modifier, Bounded, Light-tailed

The truncated normal distribution is a continuous probability distribution that is a normal distribution restricted to a specific range. It is defined by four parameters: the mean (\(\mu\)), the standard deviation (\(\sigma\)), and the lower and upper bounds of the range.

Truncated normal distributions are commonly used in cases where the observed data is known to fall within a certain range due to physical constraints or measurement limitations.

Key properties and parameters:#

Support

\(x \in [lower, upper]\)

Mean

\(\mu +{\frac {\phi (\alpha )-\phi (\beta )}{Z}}\sigma\)

Variance

\(\sigma^2 \left[1+\frac{\alpha \phi(\alpha)-\beta \phi(\beta)}{Z}-\left(\frac{\phi(\alpha)-\phi(\beta)}{Z}\right)^2\right]\)

where:

  • \(\phi\) is the standard normal PDF

  • \(\alpha = \frac{lower-\mu}{\sigma}\)

  • \(\beta = \frac{upper-\mu}{\sigma}\)

  • \(Z = \Phi(\beta) - \Phi(\alpha)\)

  • \(\Phi\) is the standard normal CDF

Parameters:

  • \(\mu\) (float): Mean of the distribution.

  • \(\sigma\) (float): Standard deviation of the distribution.

  • lower (float): Lower bound of the range.

  • upper (float): Upper bound of the range.

Probability Density Function (PDF)#

\[\begin{split} \begin{cases} 0 & \text{for } x < \text{lower}, \\ \frac{\phi\left(\frac{x-\mu}{\sigma}\right)}{\sigma(Z)} & \text{for } \text{lower} <= x <= \text{upper}, \\ 0 & \text{for } x > \text{upper}, \end{cases} \end{split}\]

where \(\phi\) is the standard normal PDF, and \(Z = \Phi(\beta) - \Phi(\alpha)\).

Truncated Normal Distribution PDF

Cumulative Distribution Function (CDF)#

\[\begin{split} \begin{cases} 0 & \text{for } x < \text{lower}, \\ \frac{\Phi\left(\frac{x-\mu}{\sigma}\right) - \Phi(\alpha)}{Z} & \text{for } \text{lower} <= x <= \text{upper}, \\ 1 & \text{for } x > \text{upper}, \end{cases} \end{split}\]

where \(\Phi\) is the standard normal CDF, and \(Z = \Phi(\beta) - \Phi(\alpha)\).

Truncated Normal Distribution CDF

See also

Related Distributions:

  • Normal Distribution - The normal distribution is an unbounded version of the truncated normal distribution.

  • Truncated Distribution - A modifier of a base distribution that restricts the values to a specific range.

References#