If X is a discrete random variable (on some probability space) and g : R → R, then Y = g(X) is a discrete random variable also. According to the above definition, we need to know the mass function of Y before we can calculate its expectation. The following theorem provides a useful way of avoiding this tedious calculation.

Theorem 2.29 (Law of the subconscious statistician) If X is a discrete random variable and g : R → R, then E(g(X)) =

X x∈Im X g(x)P(X = x),