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.
Definition The following discussion is in terms of a random variable having a continuous distribution although the same ideas apply to discrete distributions. Similarly, the discussion assumes that truncation is to a semi-open interval y ∈ (a,b] but other possibilities can be handled straightforwardly. Suppose we have a random variable, X {\displaystyle X} that is distributed according to some probability density function, f ( x ) {\displaystyle f(x)} , with cumulative distribution function F ( x ) {\displaystyle F(x)} both of which have infinite support. Suppose we wish to know the probability density of the random variable after restricting the support to be between two constants so that the support, y = ( a , b ] {\displaystyle y=(a,b]} . That is to say, suppose we wish to know how X {\displaystyle X} is distributed given a < X ≤ b {\displaystyle a<X\leq b} .
f ( x | a < X ≤ b ) = g ( x ) F ( b ) − F ( a ) = f ( x ) ⋅ I ( { a < x ≤ b } ) F ( b ) − F ( a ) ∝ x f ( x ) ⋅ I ( { a < x ≤ b } ) {\displaystyle f(x|a<X\leq b)={\frac {g(x)}{F(b)-F(a)}}={\frac {f(x)\cdot I(\{a<x\leq b\})}{F(b)-F(a)}}\propto _{x}f(x)\cdot I(\{a<x\leq b\})}
where g ( x ) = f ( x ) {\displaystyle g(x)=f(x)} for all a < x ≤ b {\displaystyle a<x\leq b} and g ( x ) = 0 {\displaystyle g(x)=0} everywhere else. That is, g ( x ) = f ( x ) ⋅ I ( { a < x ≤ b } ) {\displaystyle g(x)=f(x)\cdot I(\{a<x\leq b\})} where I {\displaystyle I} is the indicator function. Note that the denominator in the truncated distribution is constant with respect to the x {\displaystyle x} . Notice that in fact f ( x | a < X ≤ b ) {\displaystyle f(x|a<X\leq b)} is a density:
∫ a b f ( x | a < X ≤ b ) d x = 1 F ( b ) − F ( a ) ∫ a b g ( x ) d x = 1 {\displaystyle \int _{a}^{b}f(x|a<X\leq b)dx={\frac {1}{F(b)-F(a)}}\int _{a}^{b}g(x)dx=1} . Truncated distributions need not have parts removed from the top and bottom. A truncated distribution where just the bottom of the distribution has been removed is as follows:
f ( x | X > y ) = g ( x ) 1 − F ( y ) {\displaystyle f(x|X>y)={\frac {g(x)}{1-F(y)}}}
… excerpt ends here. Continue reading the full article.
