and the expectation of X is often called the expected value or mean of X.3 The reason for requiring absolute convergence in (2.28) is that the image Im X may be an infinite set, and we 3One should be careful to avoid ambiguity in the use (or not) of parentheses. For example, we shall sometimes write E(X)2 for [E(X)]2, and E|X| for E(|X|).

2.4 Expectation 31 need the summation in (2.28) to take the same value irrespective of the order in which we add up its terms.

The physical analogy of ‘expectation’ is the idea of ‘centre of gravity’. If masses with weights π1, π2, . . . are placed at the points x1, x2, . . . of R, then the position of the centre of gravity is P

πi xi

P

πi , or P

xi pi , where pi = πi

P j πj is the proportion of the total weight allocated to position xi .