The indicator function of A is denoted by 1A and defined as follows: For any event W, we have IA(W) = 1 if W belongs to A, and IA(W) = 0 if W does not belong to A.
Note that IA is a random variable. (a) Show that E[1A] = P(A) and Var[1A] = P(A)(1 - P(A)). (This correspondence allows us to perform calculations and operations using algebra on indicator functions.) (b) Show that the operations of union and intersection on sets can be translated to operations on indicator functions. Specifically, IAUB = 1 if either IA or IB is equal to 1, and IAUB = 0 otherwise. Similarly, IAnB = 1 if both IA and IB are equal to 1, and IAnB = 0 otherwise. Use this to prove the identity P(AUB) = P(A) + P(B) - P(AnB).
