Truncated distribution

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Truncated Distribution

Probability density function Probability density function for the truncated normal distribution for different sets of parameters. In all cases, a = −10 and b = 10. For the black: μ = −8, σ = 2; blue: μ = 0, σ = 2; red: μ = 9, σ = 10; orange: μ = 0, σ = 10.
Support	${\displaystyle x\in (a,b]}$
PDF	${\displaystyle {\frac {g(x)}{F(b)-F(a)}}}$
CDF	${\displaystyle {\frac {\int _{a}^{x}dF(t)}{F(b)-F(a)}}={\frac {F(x)-F(a)}{F(b)-F(a)}}}$
Mean	${\displaystyle {\frac {\int _{a}^{b}xdF(x)}{F(b)-F(a)}}}$
Median	${\displaystyle F^{-1}\left({\frac {F(a)+F(b)}{2}}\right)}$

In statistics, a truncated distribution is a conditional distribution that results from restricting the domain of some other probability distribution. Truncated distributions arise in practical statistics 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.

Where sampling is such as to retain knowledge of items that fall outside the required range, without recording the actual values, this is known as censoring, as opposed to the truncation here.^[1]