The Truncated Normal Distribution
Density, distribution function, quantile function, and random generation
for the truncated normal distribution with parameters mean,
sd, min, and max.
dnormTrunc(x, mean = 0, sd = 1, min = -Inf, max = Inf) pnormTrunc(q, mean = 0, sd = 1, min = -Inf, max = Inf) qnormTrunc(p, mean = 0, sd = 1, min = -Inf, max = Inf) rnormTrunc(n, mean = 0, sd = 1, min = -Inf, max = Inf)
x |
vector of quantiles. |
q |
vector of quantiles. |
p |
vector of probabilities between 0 and 1. |
n |
sample size. If |
mean |
vector of means of the distribution of the non-truncated random variable.
The default is |
sd |
vector of (positive) standard deviations of the non-truncated random variable.
The default is |
min |
vector of minimum values for truncation on the left. The default value is
|
max |
vector of maximum values for truncation on the right. The default value is
|
See the help file for the normal distribution for information about the density and cdf of a normal distribution.
Probability Density and Cumulative Distribution Function
Let X denote a random variable with density function f(x) and
cumulative distribution function F(x), and let
Y denote the truncated version of X where Y is truncated
below at min=A and above atmax=B. Then the density
function of Y, denoted g(y), is given by:
g(y) = frac{f(y)}{F(B) - F(A)}, A ≤ y ≤ B
and the cdf of Y, denoted G(y), is given by:
| G(y) = | 0 | for y < A |
| \frac{F(y) - F(A)}{F(B) - F(A)} | for A ≤ y ≤ B | |
| 1 | for y > B | |
Quantiles
The p^{th} quantile y_p of Y is given by:
| y_p = | A | for p = 0 |
| F^{-1}\{p[F(B) - F(A)] + F(A)\} | for 0 < p < 1 | |
| B | for p = 1 | |
Random Numbers
Random numbers are generated using the inverse transformation method:
y = G^{-1}(u)
where u is a random deviate from a uniform [0, 1] distribution.
Mean and Variance
The expected value of a truncated normal random variable with parameters
mean=μ, sd=σ, min=A, and
max=B is given by:
E(Y) = μ + σ^2 \frac{f(A) - f(B)}{F(B) - F(A)}
(Johnson et al., 1994, p.156; Schneider, 1986, p.17).
The variance of this random variable is given by:
σ^2 + σ^3 \{z_A f(A) - z_B f(B) - σ[f(A) - f(B)]^2 \}
where
z_A = \frac{A - μ}{σ}; \, z_B = \frac{B - μ}{σ}
(Johnson et al., 1994, p.158; Schneider, 1986, p.17).
dnormTrunc gives the density, pnormTrunc gives the distribution function,
qnormTrunc gives the quantile function, and rnormTrunc generates random
deviates.
A truncated normal distribution is sometimes used as an input distribution for probabilistic risk assessment.
Steven P. Millard (EnvStats@ProbStatInfo.com)
Forbes, C., M. Evans, N. Hastings, and B. Peacock. (2011). Statistical Distributions. Fourth Edition. John Wiley and Sons, Hoboken, NJ.
Johnson, N. L., S. Kotz, and N. Balakrishnan. (1994). Continuous Univariate Distributions, Volume 1. Second Edition. John Wiley and Sons, New York.
Schneider, H. (1986). Truncated and Censored Samples from Normal Populations. Marcel Dekker, New York, Chapter 2.
# Density of a truncated normal distribution with parameters # mean=10, sd=2, min=8, max=13, evaluated at 10 and 11.5: dnormTrunc(c(10, 11.5), 10, 2, 8, 13) #[1] 0.2575358 0.1943982 #---------- # The cdf of a truncated normal distribution with parameters # mean=10, sd=2, min=8, max=13, evaluated at 10 and 11.5: pnormTrunc(c(10, 11.5), 10, 2, 8, 13) #[1] 0.4407078 0.7936573 #---------- # The median of a truncated normal distribution with parameters # mean=10, sd=2, min=8, max=13: qnormTrunc(.5, 10, 2, 8, 13) #[1] 10.23074 #---------- # A random sample of 3 observations from a truncated normal distribution # with parameters mean=10, sd=2, min=8, max=13. # (Note: the call to set.seed simply allows you to reproduce this example.) set.seed(20) rnormTrunc(3, 10, 2, 8, 13) #[1] 11.975223 11.373711 9.361258
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