Truncated Normal distribution

Density, distribution function, quantile function and random generation for the truncated Normal distribution with mean equal to mean and standard deviation equal to sd before truncation, and truncated on the interval [lower, upper].

dtnorm(x, mean=0, sd=1, lower=-Inf, upper=Inf, log = FALSE)
     ptnorm(q, mean=0, sd=1, lower=-Inf, upper=Inf, lower.tail = TRUE, log.p = FALSE)
     qtnorm(p, mean=0, sd=1, lower=-Inf, upper=Inf, lower.tail = TRUE, log.p = FALSE)
     rtnorm(n, mean=0, sd=1, lower=-Inf, upper=Inf)
vector of quantiles.
vector of probabilities.
number of observations. If length(n) > 1, the length is taken to be the number required.
vector of means.
vector of standard deviations.
lower truncation point.
upper truncation point.
log, log.p
logical; if TRUE, probabilities p are given as log(p).
logical; if TRUE (default), probabilities are P[X <= x],="" otherwise,="" p[x=""> x].

The truncated normal distribution has density

$$f(x, \mu, \sigma) = \phi(x, \mu, \sigma) / (\Phi(u, \mu, \sigma) - \Phi(l, \mu, \sigma))$$ for $l <= 0="" x="" <="u$," and="" otherwise.="" p="">

$\mu$ is the mean of the original Normal distribution before truncation, $\sigma$ is the corresponding standard deviation, $u$ is the upper truncation point, $l$ is the lower truncation point, $\phi(x)$ is the density of the corresponding normal distribution, and $\Phi(x)$ is the distribution function of the corresponding normal distribution. If mean or sd are not specified they assume the default values of 0 and 1, respectively.

If lower or upper are not specified they assume the default values of -Inf and Inf, respectively, corresponding to no lower or no upper truncation. Therefore, for example, dtnorm(x), with no other arguments, is simply equivalent to dnorm(x). Only rtnorm is used in the msm package, to simulate from hidden Markov models with truncated normal distributions. These functions are merely provided for completion, and are not optimized for numerical stability. To fit a hidden Markov model with a truncated Normal response distribution, use a hmmTNorm constructor. See the hmm-dists help page for further details.


  • dtnorm gives the density, ptnorm gives the distribution function, qtnorm gives the quantile function, and rtnorm generates random deviates.

See Also


  • tnorm
  • dtnorm
  • ptnorm
  • qtnorm
  • rtnorm
x <- seq(50, 90, by=1)
plot(x, dnorm(x, 70, 10), type="l", ylim=c(0,0.06)) ## standard Normal distribution
lines(x, dtnorm(x, 70, 10, 60, 80), type="l")       ## truncated Normal distribution
Documentation reproduced from package msm, version 0.5.1, License: GPL version 2 or newer

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