# CDF for a probit model

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?

• 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 $$\varepsilon$$).