Stat 5101 Lecture Notes - School of Statistics

36 Stat 5101 (Geyer) Course NotesThese two properties are obvious consequences of linearity (Problems 2-3and 2-4).Corollary 2.7 (Constants). Every constant random variable is in L 1 , andE(a) =a.This uses the convention we introduced in connection with (2.4). The symbol“a” on the right hand side represents a real number, but the symbol “a” ontheleft hand side represents the constant random variable always equal to thatnumber. The proof is left as an exercise (Problem 2-6).Note that a special case of Corollary 2.7 is E(0) = 0.Theorem 2.8 (Monotonicity). If X and Y are in L 1 , thenX ≤ Y implies E(X) ≤ E(Y ).The expression X ≤ Y , written out in full, meansX(s) ≤ Y (s), s ∈ S,where S is the sample space. That is, X is always less than or equal to Y .Note that the positivity axiom (E3) is the special case X = 0 of this theorem.Thus this theorem is a generalization of that axiom.This theorem is fairly easily derived from the positivity axiom (E3) and theTheorem 2.5 (Problem 2-7).All of the theorems in this section and the axioms in the preceding sectionare exceedingly important and will be used continually throughout the course.You should have them all at your fingertips. Failure to recall the appropriateaxiom or theorem when required will mean failure to do many problems. It isnot necessary to memorize all the axioms and theorems. You can look themup when needed. But you do need to have some idea what each axiom andtheorem is about so you will know that there is something to look up. After all,you can’t browse the entire course notes each time you use something.Axiom E3 and Theorem 2.8 are important in what I call “sanity checks.”Suppose you are given a description of a random variable X and are told tocalculate its expectation. One of the properties given is X ≥ 3, but your answeris E(X) = 2. This is obviously wrong. It violates Theorem 2.8. You must havemade a mistake somewhere! Sanity checks like this can save you from manymistakes if you only remember to make them. A problem isn’t done when youobtain an answer. You should also take a few seconds to check that your answerisn’t obviously ridiculous.2.3.3 Important Non-PropertiesWhat’s a non-property? It’s a property that students often use but isn’ttrue. Students are mislead by analogy or guessing. Thus we stress that thefollowing are not true in general (although they are sometimes true in somespecial cases).