This preview shows pages 1–11. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Stat 5101 Lecture Slides Deck 2 Charles J. Geyer School of Statistics University of Minnesota 1 Axioms An expectation operator is a mapping X 7 E ( X ) of random variables to real numbers that satisfies the following axioms: E ( X + Y ) = E ( X ) + E ( Y ) for any random variables X and Y , E ( X ) for any nonnegative random variable X (one such that X ( s ) for all s in the sample space), E ( aX ) = aE ( X ) for any random variable X and any constant a , and E ( Y ) = 1 when Y is the constant random variable s 7 1. 2 Axioms (cont.) The fourth axiom is usually written, a bit sloppily, as E (1) = 1 The reason this is sloppy is that on the lefthand side 1 must indicate a random variable, because the argument of an expecta tion operator is always a random variable, and on the righthand side 1 must indicate a real number, because the value of an expectation operator is always a real number. When we have a constant as an argument of an expectation op erator, we always take this to mean a constant random variable. 3 Axiom Summary E ( X + Y ) = E ( X ) + E ( Y ) (1) E ( X ) , when X (2) E ( aX ) = aE ( X ) (3) E (1) = 1 (4) (3) and (4) together imply E ( a ) = a, for any constant a It can be shown (but we wont here) that when the sample space is finite these axioms hold if and only if the expectation operator is defined in terms of a PMF as we did before. 4 Axioms (cont.) E ( X + Y ) = E ( X ) + E ( Y ) says an addition operation can be pulled outside an expectation. X implies E ( X ) says nonnegativity can be pulled outside an expectation. E ( aX ) = aE ( X ) says a constant can be pulled outside an expectation. 5 Axioms (cont.) Many students are tempted to overgeneralize, and think anything can be pulled outside an expectation. Wrong! In general E ( XY ) 6 = E ( X ) E ( Y ) E ( X/Y ) 6 = E ( X ) /E ( Y ) E { g ( X ) } 6 = g E { X } although we may have equality for certain special cases. 6 Axioms (cont.) We do have E ( X Y ) = E ( X ) E ( Y ) because E ( X Y ) = E { X + ( 1) Y } = E ( X ) + ( 1) E ( Y ) by axioms (1) and (3). 7 Axioms (cont.) We do have E ( a + bX ) = a + bE ( X ) because E ( a + bX ) = E ( a ) + E ( bX ) = a + bE ( X ) by axioms (1), (3), and (4). 8 Axiom Summary (cont.) E ( X Y ) = E ( X ) E ( Y ) addition and subtraction come out X implies E ( X ) nonnegative comes out E ( aX ) = aE ( X ) constants come out E ( a + bX ) = a + bE ( X ) linear functions come out. But thats all! 9 Linearity of Expectation By mathematical induction the addition comes out axiom ex tends to any finite number of random variables. For any random variables X 1 , ... , X n E ( X 1 + + X n ) = E ( X 1 ) + + E ( X n ) More generally, for any random variables X 1 , ... , X n and any constants a 1 , ... , a n E ( a 1 X 1 + + a n X n ) = a 1 E ( X 1 ) + + a n E ( X n ) This very useful property is called linearity of expectation ....
View Full
Document
 Fall '02
 Staff
 Statistics

Click to edit the document details