Truncated Distribution#

Modifier

This is not a distribution per se, but a modifier of univariate distributions.

Truncated distributions arise in cases where the ability to record, or even to know about, occurrences is limited to values which lie above or below a given threshold or within a specified range. For example, if the dates of birth of children in a school are examined, these would typically be subject to truncation relative to those of all children in the area given that the school accepts only children in a given age range on a specific date. There would be no information about how many children in the locality had dates of birth before or after the school’s cutoff dates if only a direct approach to the school were used to obtain information.

Key properties and parameters#

Parameters:

  • dist : (PreliZ distribution) Univariate distribution to be truncated.

  • lower : (float, int, or np.inf) Lower (left) truncation point, np.inf indicates no lower truncation.

  • upper : (float, int, or np.inf) Upper (right) truncation point, np.inf indicates no upper truncation.

Probability Density Function (PDF)#

Given a base distribution with cumulative distribution function (CDF) and probability density mass/function (PDF). The pdf of a Truncated distribution is:

\[\begin{split} \begin{cases} 0 & \text{for } x < \text{lower}, \\ \frac{\text{PDF}(x)}{\text{CDF}(upper) - \text{CDF}(lower)} & \text{for } \text{lower} <= x <= \text{upper}, \\ 0 & \text{for } x > \text{upper}, \end{cases} \end{split}\]

where lower and upper are the lower and upper bounds of the truncated distribution, respectively.

Truncated Distribution PDF

Cumulative Distribution Function (CDF)#

The given expression can be written mathematically as:

\[\begin{split} \begin{cases} 0 & \text{if } x_i < \text{lower} \\ 1 & \text{if } x_i > \text{upper} \\ \frac{\text{CDF}(x_i) - \text{CDF}(\text{lower})}{\text{CDF}(\text{upper}) - \text{CDF}(\text{lower})} & \text{if } \text{lower} \leq x_i \leq \text{upper} \end{cases} \end{split}\]

where lower and upper are the lower and upper bounds of the truncated distribution, respectively.

Trucated Distribution CDF

See also

Related Distributions:

  • Censored - In a censored distribution, values outside the range are set to the nearest bound, while in a truncated distribution, they are not recorded.

  • TruncatedNormal - A truncated normal distribution is a normal distribution that has been restricted to a specific range.

References#