Indicator variables and Bernouilli variables

An indicator variable for the event A is defined as the random variable that takes on only 2 values 0 and 1, it takes 1 when event A happens and 0 otherwise. So that the expectation of this indicator(noted I_A) is

$$E(I_A)=0\times P(A^c)+ 1 \times P(A)=P(A)$$

. Very often we just define an event ``success'' and we are told that success happens with probability p, the random variable X that takes on 1 with probability p and 0 with probability 1-p is called the Bernouilli(p) random variable.

E(X)=p