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. For example, it is used to model the probabilities of the binary outcomes in the probit model and to model censored data in the tobit model.

Probability density function
Probability density function for the truncated normal distribution for different sets of parameters. In all cases, a = −10 and b = 10. For the black: μ = −8, σ = 2; blue: μ = 0, σ = 2; red: μ = 9, σ = 10; orange: μ = 0, σ = 10.
Cumulative distribution function
Cumulative distribution function for the truncated normal distribution for different sets of parameters. In all cases, a = −10 and b = 10. For the black: μ = −8, σ = 2; blue: μ = 0, σ = 2; red: μ = 9, σ = 10; orange: μ = 0, σ = 10.
Parameters μR
σ2 ≥ 0 (but see definition)
a ∈ R — minimum value of x
b ∈ R — maximum value of x (b > a)
Support x ∈ [a,b]
PDF [1]


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.



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 canonical 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 a canonical 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].


If the random variable has been truncated only from below, some probability mass has been shifted to higher values, giving a first-order stochastically dominating distribution and hence increasing the mean to a value higher than the mean   of the original normal distribution. Likewise, if the random variable has been truncated only from above, the truncated distribution has a mean less than  

Regardless of whether the random variable is bounded above, below, or both, the truncation is a mean-preserving contraction combined with a mean-changing rigid shift, and hence the variance of the truncated distribution is less than the variance   of the original normal distribution.

Two sided truncation[2]Edit

Let   and  . Then:




Care must be taken in the numerical evaluation of these formulas, which can result in catastrophic cancellation when the interval   does not include  . There are better ways to rewrite them that avoid this issue.[3]

One sided truncation (of lower tail)[4]Edit

In this case   then





One sided truncation (of upper tail)Edit

In this case   then



Barr and Sherrill (1999) give a simpler expression for the variance of one sided truncations. Their formula is in terms of the chi-square CDF, which is implemented in standard software libraries. Bebu and Mathew (2009) provide formulas for (generalized) confidence intervals around the truncated moments.

A recursive formula

As for the non-truncated case, there is a recursive formula for the truncated moments.[5]


Computing the moments of a multivariate truncated normal is harder.

Computational methodsEdit

Generating values from the truncated normal distributionEdit

A random variate x defined as   with   the cumulative distribution function and   its inverse,   a uniform random number on  , follows the distribution truncated to the range  . This is simply the inverse transform method for simulating random variables. Although one of the simplest, this method can either fail when sampling in the tail of the normal distribution,[6] or be much too slow.[7] Thus, in practice, one has to find alternative methods of simulation.

One such truncated normal generator (implemented in Matlab and in R (programming language) as trandn.R ) is based on an acceptance rejection idea due to Marsaglia.[8] Despite the slightly suboptimal acceptance rate of Marsaglia (1964) in comparison with Robert (1995), Marsaglia's method is typically faster,[7] because it does not require the costly numerical evaluation of the exponential function.

For more on simulating a draw from the truncated normal distribution, see Robert (1995), Lynch (2007) Section 8.1.3 (pages 200–206), Devroye (1986). The MSM package in R has a function, rtnorm, that calculates draws from a truncated normal. The truncnorm package in R also has functions to draw from a truncated normal.

Chopin (2011) proposed (arXiv) an algorithm inspired from the Ziggurat algorithm of Marsaglia and Tsang (1984, 2000), which is usually considered as the fastest Gaussian sampler, and is also very close to Ahrens’s algorithm (1995). Implementations can be found in C, C++, Matlab and Python.

Sampling from the multivariate truncated normal distribution is considerably more difficult.[9] Exact or perfect simulation is only feasible in the case of truncation of the normal distribution to a polytope region.[9][10] In more general cases, Damien and Walker (2001) introduce a general methodology for sampling truncated densities within a Gibbs sampling framework. Their algorithm introduces one latent variable and, within a Gibbs sampling framework, it is more computationally efficient than the algorithm of Robert (1995).

See alsoEdit


  1. ^ "Lecture 4: Selection" (PDF). Instituto Superior Técnico. November 11, 2002. p. 1. Retrieved 14 July 2015.
  2. ^ Johnson, N.L., Kotz, S., Balakrishnan, N. (1994) Continuous Univariate Distributions, Volume 1, Wiley. ISBN 0-471-58495-9 (Section 10.1)
  3. ^ Fernandez-de-Cossio-Diaz, Jorge (2017-12-06), TruncatedNormal.jl: Compute mean and variance of the univariate truncated normal distribution (works far from the peak), retrieved 2017-12-06
  4. ^ Greene, William H. (2003). Econometric Analysis (5th ed.). Prentice Hall. ISBN 978-0-13-066189-0.
  5. ^ Document by Eric Orjebin, ""
  6. ^ Kroese, D. P.; Taimre, T.; Botev, Z. I. (2011). Handbook of Monte Carlo methods. John Wiley & Sons.
  7. ^ a b Botev, Z. I.; L'Ecuyer, P. (2017). "Simulation from the Normal Distribution Truncated to an Interval in the Tail". 10th EAI International Conference on Performance Evaluation Methodologies and Tools. 25th–28th Oct 2016 Taormina, Italy: ACM. pp. 23–29. doi:10.4108/eai.25-10-2016.2266879. ISBN 978-1-63190-141-6.CS1 maint: location (link)
  8. ^ Marsaglia, George (1964). "Generating a variable from the tail of the normal distribution". Technometrics. 6 (1): 101–102. doi:10.2307/1266749. JSTOR 1266749.
  9. ^ a b Botev, Z. I. (2016). "The normal law under linear restrictions: simulation and estimation via minimax tilting". Journal of the Royal Statistical Society, Series B. 79: 125–148. arXiv:1603.04166. doi:10.1111/rssb.12162. S2CID 88515228.
  10. ^ Botev, Zdravko & L'Ecuyer, Pierre (2018). "Chapter 8: Simulation from the Tail of the Univariate and Multivariate Normal Distribution". In Puliafito, Antonio (ed.). Systems Modeling: Methodologies and Tools. EAI/Springer Innovations in Communication and Computing. Springer, Cham. pp. 115–132. doi:10.1007/978-3-319-92378-9_8. ISBN 978-3-319-92377-2. S2CID 125554530.