Truncated normal distribution

In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated normal distribution has wide applications in statistics and econometrics. Suppose has a normal distribution with mean and variance and lies within the interval . Then conditional on has a truncated normal distribution. Its probability density function, , for , is given by and by otherwise. Here, is the probability density function of the standard normal distribution and is its cumulative distribution function By definition, if , then , and similarly, if , then . The above formulae show that when the scale parameter of the truncated normal distribution is allowed to assume negative values. The parameter is in this case imaginary, but the function is nevertheless real, positive, and normalizable. The scale parameter of the untruncated normal distribution must be positive because the distribution would not be normalizable otherwise. The doubly truncated normal distribution, on the other hand, can in principle have a negative scale parameter (which is different from the variance, see summary formulae), because no such integrability problems arise on a bounded domain. In this case the distribution cannot be interpreted as an untruncated normal conditional on , of course, but can still be interpreted as a maximum-entropy distribution with first and second moments as constraints, and has an additional peculiar feature: it presents two local maxima instead of one, located at and . The truncated normal is the maximum entropy probability distribution for a fixed mean and variance with the random variate X constrained to be in the interval [a,b]. Truncated normals with fixed support form an exponential family. Nielsen reported closed-form formula for calculating the Kullback-Leibler divergence and the Bhattacharyya distance between two truncated normal distributions with the support of the first distribution nested into the support of the second distribution.