Truncated normal distribution

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.
Notation	$\xi ={\frac {x-\mu }{\sigma }},\ \alpha ={\frac {a-\mu }{\sigma }},\ \beta ={\frac {b-\mu }{\sigma }}$ $\xi ={\frac {x-\mu }{\sigma }},\ \alpha ={\frac {a-\mu }{\sigma }},\ \beta ={\frac {b-\mu }{\sigma }}$ $Z=\Phi (\beta )-\Phi (\alpha )$ $Z=\Phi (\beta )-\Phi (\alpha )$
Parameters	μ ∈ R — location σ² ≥ 0 — squared scale a ∈ R — minimum value b ∈ R — maximum value
Support	x ∈ [a,b]
PDF	$f(x;\mu ,\sigma ,a,b)={\frac {\phi (\xi )}{\sigma Z}}\,$ $f(x;\mu ,\sigma ,a,b)={\frac {\phi (\xi )}{\sigma Z}}\,$
CDF	$F(x;\mu ,\sigma ,a,b)={\frac {\Phi (\xi )-\Phi (\alpha )}{Z}}$ $F(x;\mu ,\sigma ,a,b)={\frac {\Phi (\xi )-\Phi (\alpha )}{Z}}$
Mean	$\mu +{\frac {\phi (\alpha )-\phi (\beta )}{Z}}\sigma$ $\mu +{\frac {\phi (\alpha )-\phi (\beta )}{Z}}\sigma$
Mode	$\left\{{\begin{array}{ll}a,&\mathrm {if} \ \mu <a\\\mu ,&\mathrm {if} \ a\leq \mu \leq b\\b,&\mathrm {if} \ \mu >b\end{array}}\right.$ $\left\{{\begin{array}{ll}a,&\mathrm {if} \ \mu <a\\\mu ,&\mathrm {if} \ a\leq \mu \leq b\\b,&\mathrm {if} \ \mu >b\end{array}}\right.$
Variance	$\sigma ^{2}\left[1+{\frac {\alpha \phi (\alpha )-\beta \phi (\beta )}{Z}}-\left({\frac {\phi (\alpha )-\phi (\beta )}{Z}}\right)^{2}\right]$ $\sigma ^{2}\left[1+{\frac {\alpha \phi (\alpha )-\beta \phi (\beta )}{Z}}-\left({\frac {\phi (\alpha )-\phi (\beta )}{Z}}\right)^{2}\right]$
Entropy	$\ln({\sqrt {2\pi e}}\sigma Z)+{\frac {\alpha \phi (\alpha )-\beta \phi (\beta )}{2Z}}$ $\ln({\sqrt {2\pi e}}\sigma Z)+{\frac {\alpha \phi (\alpha )-\beta \phi (\beta )}{2Z}}$
MGF	$e^{\mu t+\sigma ^{2}t^{2}/2}\left[{\frac {\Phi (\beta -\sigma t)-\Phi (\alpha -\sigma t)}{\Phi (\beta )-\Phi (\alpha )}}\right]$ $e^{\mu t+\sigma ^{2}t^{2}/2}\left[{\frac {\Phi (\beta -\sigma t)-\Phi (\alpha -\sigma t)}{\Phi (\beta )-\Phi (\alpha )}}\right]$

In probability and statistics, the truncated normal distribution is the probability distribution of a normally distributed random variable whose value is either bounded 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.

Suppose $X\sim N(\mu ,\sigma ^{2})$ $X\sim N(\mu ,\sigma ^{2})$ has a normal distribution and lies within the interval $X\in (a,b),\;-\infty \leq a<b\leq \infty$ $X\in (a,b),\;-\infty \leq a<b\leq \infty$ . Then $X$ $X$ conditional on $a<X<b$ $a<X<b$ has a truncated normal distribution.

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