The cdf for a probit model is:

$$ \Phi(\varepsilon)=\int_{-\infty}^\varepsilon \frac{1}{\sqrt{2\pi}} \exp\left(-\frac{t^{2}}{2}\right) \, dt $$

My very simply question—that I should know the answer to but apparently don't—is what's $d$ and $t$ in the formula?


Your question reveal some confusion about generalized linear models and calculus:

  • Concerning the terminology: the one you listed is the cdf (cumulative distribution function) of the standard normal distribution; a probit model is not a probability distribution so it doesn't have a cdf.
  • However probit models are characterized by a link function (as all generalized linear models) that transform the linear part of the model into a probability (values between 0 and 1), which happens to be the cdf of the standard normal distribution.
  • The cdf of a random variable $X$, evaluated at $x$, is the probability of $X$ taking a values less or equal than $x$ (see wikipedia page). For a continuous random variable is calculated as the integral of the probability density function. In fact in your formula within the integral you have the probability density function of the random variable $t$; the formula indicate that it has a standard normal distribution. $dt$ indicates the differential of the variable $t$, and is there to indicate that the integral is taken with respect to $t$ (essentially, the formula expresses the probability that the variable $t$ takes a value less or equal than $\varepsilon$).
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