Expectation-Examples

Abstract

The general theory of expectation was outlined in Lessons 11 and 12 Part II. In this lesson, we spell out more particular properties of expectation needed later. For simplicity, we assume that all RVs are defined on a given probability space [ΩβP] and, unless otherwise indicated, that all the expectations are finite. The basic form is:

• X: Ω → Rⁿis measurable wrt (β β _n) and has CDF FX;

• g: X(Ω) → R is measurable wrt (β _n β ₁);

• $$E[g(X)] = \int_{\Omega } {g(X(\omega ))dP(\omega ) = \int_{{{R^{n}}}} {g(x) d{F_{X}}(x).} }$$

The following particular cases are used in practice.

• a) For n=1 with a < b, both finite such that P (a ≤ X ≤ b) =1

  $$E[g(X)] = \int_{a}^{b} {g(x)d{F_{X}}(x).}$$

• b) If n = 1 but X is unbounded

  $$E[g(X)] = \mathop{{\lim }}\limits_{{a,b \to \infty }} \int_{{ - a}}^{b} {g(x)d{F_{X}}(x).}$$

• c) If n = 1 and X (Ω) can be well-ordered as x₀< x₁< …, at most countable

  $$\begin{array}{*{20}{c}} {E[g(x)] = \sum\nolimits_{{i = 0}}^{\infty } {g({x_{i}})({F_{X}}({x_{i}}) - {F_{X}}({x_{{i - 1}}}))} } \\ { = \sum\nolimits_{{i = 0}}^{\infty } {g({x_{i}})P(X = {x_{i}}).} } \\ \end{array}$$

• d) If the support of X is finite like^{y₁y₂…, y_m)} (for any n ≥ 1), E[g(X)] is the finite sum

  g(y₁)P(X = y₁) +… + g(Y_m)P(X = y_m).

  $$g({y_{1}})P(X = {y_{1}}) + \cdots + g({y_{m}})P(X = {y_{m}}).$$

• e) If g is continuous and the density f_Xis continuous a.e., then