Stat 5101 Lecture Notes - School of Statistics

2.3. BASIC PROPERTIES 35that is, the expectations transform the same way as the variables under a changeof units. Thus, if the expected daily high temperature in January in Minneapolisis 23 ◦ F, then this expected value is also −5 ◦ C. Expectations behave sensiblyunder changes of units of measurement.Theorem 2.3 (Linearity). If X 1 , ..., X n are in L 1 , and a 1 , ..., a n are realnumbers then a 1 X 1 + ···a n X n is also in L 1 , andE(a 1 X 1 + ···+a n X n )=a 1 E(X 1 )+···+a n E(X n ).Theorem 2.1 is the case n = 2 of Theorem 2.3, so the latter is a generalizationof the former. That’s why both have the same name. (If this isn’t obvious, youneed to think more about “mathematics is invariant under changes of notation.”The two theorems use different notation, a 1 and a 2 instead of a and b and X 1and X 2 instead of X and Y , but they assert the same property of expectation.)Proof of Theorem 2.3. The proof is by mathematical induction. The theoremis true for the case n = 2 (Theorem 2.1). Thus we only need to show that thetruth of the theorem for the case n = k implies the truth of the theorem for thecase n = k + 1. Apply Axiom E1 to the case n = k + 1 givingE(a 1 X 1 + ···+a k+1 X k+1 )=E(a 1 X 1 +···+a k X k )+E(a k+1 X k+1 ).Then apply Axiom E2 to the second term on the right hand side givingE(a 1 X 1 + ···+a k+1 X k+1 )=E(a 1 X 1 +···+a k X k )+a k+1 E(X k+1 ).Now the n = k case of the theorem applied to the first term on the right handside gives the n = k + 1 case of the theorem.Corollary 2.4 (Additivity). If X 1 , ..., X n are in L 1 , then X 1 + ···X n isalso in L 1 , andE(X 1 + ···+X n )=E(X 1 )+···+E(X n ).This theorem is used so often that it seems worth restating in words to helpyou remember.The expectation of a sum is the sum of the expectations.Note that Axiom E1 is the case n = 2, so the property asserted by this theoremis a generalization. It can be derived from Axiom E1 by mathematical inductionor from Theorem 2.3 (Problem 2-2).Corollary 2.5 (Subtraction). If X and Y are in L 1 , then X − Y is also inL 1 , andE(X − Y )=E(X)−E(Y).Corollary 2.6 (Minus Signs). If X is in L 1 , then −X is also in L 1 , andE(−X) =−E(X).