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.

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	${\displaystyle \xi ={\frac {x-\mu }{\sigma }},\ \alpha ={\frac {a-\mu }{\sigma }},\ \beta ={\frac {b-\mu }{\sigma }}}$ ${\displaystyle Z=\Phi (\beta )-\Phi (\alpha )}$
Parameters	${\displaystyle \mu \in \mathbb {R} }$ ${\displaystyle \sigma ^{2}\geq 0}$ (but see definition) ${\displaystyle a\in \mathbb {R} }$ — minimum value of ${\displaystyle x}$ ${\displaystyle b\in \mathbb {R} }$ — maximum value of ${\displaystyle x}$ (${\displaystyle b>a}$)
Support	${\displaystyle x\in [a,b]}$
PDF	${\displaystyle f(x;\mu ,\sigma ,a,b)={\frac {\varphi (\xi )}{\sigma Z}}\,}$
CDF	${\displaystyle F(x;\mu ,\sigma ,a,b)={\frac {\Phi (\xi )-\Phi (\alpha )}{Z}}}$
Mean	${\displaystyle \mu +{\frac {\varphi (\alpha )-\varphi (\beta )}{Z}}\sigma }$
Median	${\displaystyle \mu +\Phi ^{-1}\left({\frac {\Phi (\alpha )+\Phi (\beta )}{2}}\right)\sigma }$
Mode	${\displaystyle \left\{{\begin{array}{ll}a,&\mathrm {if} \ \mu <a\\\mu ,&\mathrm {if} \ a\leq \mu \leq b\\b,&\mathrm {if} \ \mu >b\end{array}}\right.}$
Variance	${\displaystyle \sigma ^{2}\left[1-{\frac {\beta \varphi (\beta )-\alpha \varphi (\alpha )}{Z}}-\left({\frac {\varphi (\alpha )-\varphi (\beta )}{Z}}\right)^{2}\right]}$
Entropy	${\displaystyle \ln({\sqrt {2\pi e}}\sigma Z)+{\frac {\alpha \varphi (\alpha )-\beta \varphi (\beta )}{2Z}}}$
MGF	${\displaystyle e^{\mu t+\sigma ^{2}t^{2}/2}\left[{\frac {\Phi (\beta -\sigma t)-\Phi (\alpha -\sigma t)}{\Phi (\beta )-\Phi (\alpha )}}\right]}$