Probit

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In probability theory and statistics, the probit function is the quantile function associated with the standard normal distribution. It has applications in data analysis and machine learning, in particular exploratory statistical graphics and specialized regression modeling of binary response variables.

Mathematically, the probit is the inverse of the cumulative distribution function of the standard normal distribution, which is denoted as ${\displaystyle \Phi (z)}$, so the probit is defined as

${\displaystyle \operatorname {probit} (p)=\Phi ^{-1}(p)\quad {\text{for}}\quad p\in (0,1)}$.

Largely because of the central limit theorem, the standard normal distribution plays a fundamental role in probability theory and statistics. If we consider the familiar fact that the standard normal distribution places 95% of probability between −1.96 and 1.96 and is symmetric around zero, it follows that

${\displaystyle \Phi (-1.96)=0.025=1-\Phi (1.96).\,\!}$

The probit function gives the 'inverse' computation, generating a value of a standard normal random variable, associated with specified cumulative probability. Continuing the example,

${\displaystyle \operatorname {probit} (0.025)=-1.96=-\operatorname {probit} (0.975)}$.

In general,

${\displaystyle \Phi (\operatorname {probit} (p))=p}$

and

${\displaystyle \operatorname {probit} (\Phi (z))=z.}$