# Truncated Normal Distribution

September 3, 2009
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(This article was first published on Quantitative Ecology, and kindly contributed to R-bloggers)

Many distributions may be used to describe patterns that are non-negative; however, there are not as many choices when an upper bound is also needed (although the beta distribution is very flexible). For various reasons, truncated distributions are sometimes preferred, and the truncated normal is particularly popular. While R has a package that includes the standard functions for this distribution (see rtnorm, dtnorm, etc. in the msm pacakge), the true expectation and variance of the distribution may be of interest. It turns out that the first two moments of the truncated normal are not too hard to calculate (but worth writing functions for):
mean.tnorm<-function(mu,sd,lower,upper){  ##return the expectation of a truncated normal distribution  lower.std=(lower-mu)/sd  upper.std=(upper-mu)/sd  mean=mu+sd*(dnorm(lower.std)-dnorm(upper.std))/      (pnorm(upper.std)-pnorm(lower.std))  return(mean)}var.tnorm<-function(mu,sd,lower,upper){  ##return the variance of a truncated normal distribution  lower.std=(lower-mu)/sd  upper.std=(upper-mu)/sd  variance=sd^2*(1+(lower.std*dnorm(lower.std)-upper.std*dnorm(upper.std))/    (pnorm(upper.std)-pnorm(lower.std))-((dnorm(lower.std)-dnorm(upper.std))/    (pnorm(upper.std)-pnorm(lower.std)))^2)  return(variance)}###Testing> library(msm)> a=rtnorm(1000000,-5,2,1,3)> paste(mean(a),var(a))[1] "1.52135857341077 0.197281057170982"> paste(mean.tnorm(-5,2,1,3),var.tnorm(-5,2,1,3))[1] "1.52090857118 0.197111175109889"