If you start dealing with Generalized linear models (GLMs) you will come across sentences like *“Obviously the variance of the binary dependent variable is .”* Well, for everybody who does not find it too obvious the following derivation may help in understanding the mathematical reasoning behind GLMs, especially Logit and Probit models.

Assume a binary random variable :

The relation holds since the probabilities (of a discrete random variable) must sum to 1.

The Variance of a random variable is defined as

.

The expected value of our binary random variable is

.

therefore has the nice interpretation of being the probabilty of X taking on the value 1.

With that information we can derive the variance of a binary random variate:

holds because X can only take on the values zero or one and it holds that and .

DeeptiThis was useful. thanks.