Stat 5101 Lecture Notes - School of Statistics

178 Stat 5101 (Geyer) Course NotesBy definition, the probability function of X isf(x) =P(X=x)= ∑ 1n =card({ i ∈ S : x i = x })ni∈Sx i=xwhere, as usual, card(A) denotes the cardinality of the set A. If all of the x iare distinct, then the distribution of X is also uniform. Otherwise, it is not.If the point x occurs m times among the x i , then f(x) =m/n. This makesthe definition of the empirical distribution in terms of its probability functionrather messy. So we won’t use it.The description in terms of expectation is much simpler.Definition 7.1.1 (Empirical Expectation).The empirical expectation operator associated with the vector (x 1 ,...,x n ) isdenoted E n and defined byE n {g(X)} = 1 n∑g(x i ). (7.2)nExample 7.1.2.For the data in Example 7.1.1 we have for the function g(x) =xE n (X)= 1 n∑x i =0.813nand for the function g(x) =x 2E n (X 2 )= 1 ni=1i=1n∑x 2 i =1.37819i=1The corresponding probability measure P n is found by using “probability isjust expectation of indicator functions.”Definition 7.1.2 (Empirical Probability Measure).The empirical probability measure associated with the vector (x 1 ,...,x n ) is denotedP n and defined byP n (A) = 1 n∑I A (x i ). (7.3)nExample 7.1.3.For the data in Example 7.1.1 we have for the event X>2P n (X>2) = 1 n∑I (2,∞) (x i )= number of x i greater than 2=0.1nni=1and for the event 1