First of all thanks to David Cox for popularizing such a dominating method for the analysis of binary data.

In case of linear regression, the dependent variable is a linear combination of the independent variables.

Logistic regression equation is just rewriting the linear regression equation with dependent variable enclosed in a link function.

In the above case, the link function is a logit link function.

The logit for a number p between 0 and 1 is given as below :

The link function can be defined as g(Y) = log( p(Y) / ( 1 - p(Y) ))

Since the dependent variable Y is categorical, p(Y) is simply the probability of Y belonging to different classes.

Why logit link function ?

The dependent variable Y in case of logistic regression is categorical. For simplicity lets assume it can take two values , either 0 or 1.

So its quite reasonable to be concerned about the probability of Y taking the value of either 0 or 1, i.e, P(Y) which must satisfy the following criteria

0 <= P(Y) <= 1

How can we determine the value of P(Y) so that it must always be greater than or equal to zero and less than or equal to one.

The simplest way of doing this is P(Y) = Z/(1+Z) , where Z is a positive integer.

But rather than choosing just any positive integer as Z , we choose Z = e^Y for following reasons.

(i) Bounded between 0 and 1.
(ii) Derivative can be easily calculated.
(iii) It easily introduces non-linearity into the model.