# How to compute the pdf for logit/probit models?

According to the probit/logit models, the change in probability due to a change in an explicative variable x is given by the following equation:

P(Y = 1 |X) = g($\beta{_0}$+$\beta{_1}$$x{_1}+...+\beta{_k}$$x{_k}$)$\beta{_j}$;

where g() is the probability density function underlying the model (Normal distribution in the Probit model case, while the logistic distribution in the logit model case).

Now, according to Woolridge (2009), in the case of the probit model, the value of g(0) is given by:

$\phi(0)$ = $\sqrt\frac{1}{2\pi}$ $\approx$ 0.40;

Of course, by replacing in the normal pdf the value of $X = 0$, you get that result.

My question is why one gets:

$g(0)$ $\approx$ 0.25;

by replacing the 0 value in the logistic pdf?

## 1 Answer

Be careful not to confuse the probability density function and the cumulative function. The integral of the logistic function itself is infinite, hence the logistic function can't be a PDF.

What you probably want is setting g(x) equal to the the derivative of the logistic function (i.e. the logistic function is the cumulative density function). If $f(x)$ is the logistic function, then the derivative $g(x)$ is $f(x) \cdot (1-f(x))$. Since $f(x) = 0.5$ you'll get $g(x) = 0.25$.

• Yes, it is exactly what I wanted to know. Thanks a lot for the answer. – Quantopik Feb 17 '15 at 11:17