Truncated distribution

From Wikipedia, the free encyclopedia

A truncated distribution is a conditional distribution that conditions on the random variable in question. It is derived from some other probability distribution

$f(X|X>y) = \frac{f(x)}{1-F(y)}$

where $f (x)$ is a continuous distribution function where we have redefined the support from $(-\infty, \infty)$ to $(y, \infty)$ and $F (x)$ is the cumulative distribution function with normal support, both of which are assumed to be known. In this example, we are looking at a distribution where the the bottom has been lobbed off. A truncated distribution where the bottom has been removed is as follows:

$f(X|X \leq y) = \frac{f(x)}{F(y)}$

where $f (x)$ is a continuous distribution function where we have redefined the support from $(-\infty, \infty)$ to $(\infty, y)$ and $F (x)$ is the cumulative distribution function with normal support, both of which are assumed to be known.

Notice that $\int_{y}^\infty f(X|X>y)dx = \frac{1}{1-F(y)} \int_{y}^\infty f(x) = 1$ .

The expectation of a truncated random variable is thus:

$E(X|X>y) = \frac{\int_y^\infty x f(x) dx}{1 - F(y)}$

Letting $a$ and $b$ be the lower and upper limits respectively of support for $f (x)$ properties of $E (u (X) | X > y)$ where $u (X)$ is some continuous function of $X$ with a continuous derivative and where $f (x)$ is assumed continuous include: