Stat 5101 Lecture Notes - School of Statistics

2.3. BASIC PROPERTIES 33expectation more complicated is that not all real-valued random variables haveexpectations. The set of real valued random variables that have expectation isdenoted L 1 or sometimes L 1 (P ) where P is the probability measure associatedwith the expectation, the letter “L” here being chosen in honor of the Frenchmathematician Henri Lebesgue (1875–1941), who invented the general definitionof integration used in advanced probability theory (p. 67 of these notes),the digit “1” being chosen for a reason to be explained later. The connectionbetween integration and expectation will also be explained later.An expectation operator is a function that assigns to each random variableX ∈ L 1 a real number E(X) called the expectation or expected value of X.Every expectation operator satisfies the following axioms.Axiom E1 (Additivity). If X and Y are in L 1 , then X + Y is also in L 1 ,andE(X + Y )=E(X)+E(Y).Axiom E2 (Homogeneity). If X is in L 1 and a is a real number, then aXis also in L 1 , andE(aX) =aE(X).These properties agree with either of the informal intuitions about expectations.Prices are additive and homogeneous. The price of a gallon of milk anda box of cereal is the sum of the prices of the two items separately. Also theprice of three boxes of cereal is three times the price of one box. (The notion ofexpectation as fair price doesn’t allow for volume discounts.)Axiom E3 (Positivity). If X is in L 1 , thenX ≥ 0 implies E(X) ≥ 0.The expression X ≥ 0, written out in more detail, meansX(s) ≥ 0, s ∈ S,where S is the sample space. That is, X is always nonnegative.This axiom corresponds to intuition about prices, since goods always havenonnegative value and prices are also nonnegative.Axiom E4 (Norm). The constant random variable I that always has the valueone is in L 1 , andE(I) =1. (2.3)Equation (2.3) is more commonly writtenE(1) = 1, (2.4)and we will henceforth write it this way. This is something of an abuse of notation.The symbol “1” on the right hand side is the number one, but the symbol“1” on the left hand side must be a random variable (because the argument ofan expectation operator is a random variable), hence a function on the samplespace. So in order to understand (2.4) we must agree to interpret a number in acontext that requires a random variable as the constant random variable alwaysequal to that number.