Stat 5101 Lecture Notes - School of Statistics

7.1. EMPIRICAL DISTRIBUTIONS 1797.1.1 The Mean of the Empirical DistributionFor the rest of this section we consider the special case in which all of thex i are real numbers.The mean of the empirical distribution is conventionally denoted by ¯x n andis obtained by taking the case g(x) =xin (7.2)¯x n = E n (X) = 1 n∑x i .n7.1.2 The Variance of the Empirical DistributionThe variance of the empirical distribution has no conventional notation, butwe will use both var n (X) and v n . Just like any other variance, it is the expectedsquared deviation from the mean. The mean is ¯x n ,sov n =var n (X)=E n {(X−¯x n ) 2 }= 1 n∑(x i − ¯x n ) 2 (7.4)nIt is important that you think of the empirical distribution as a probabilitydistribution just like any other. This gives us many facts about empirical distributions,that are derived from general facts about probability and expectation.For example, the parallel axis theorem holds, just as it does for any probabilitydistribution. For ease of comparison, we repeat the general parallel axis theorem(Theorem 2.11 of Chapter 2.27 of these notes).If X is a real-valued random variable having finite variance and a is any realnumber, thenE{(X − a) 2 } = var(X)+[a−E(X)] 2 (7.5)Corollary 7.1 (Empirical Parallel Axis Theorem).E n {(X − a) 2 } =var n (X)+[a−E n (X)] 2or, in other notation,1ni=1i=1n∑(x i − a) 2 = v n +(a−¯x n ) 2 (7.6)i=1In particular, the case a = 0 gives the empirical version ofvar(X) =E(X 2 )−E(X) 2which isor, in other notation,var n (X)=E n (X 2 )−E n (X) 2v n = 1 nn∑x 2 i − ¯x 2 n. (7.7)i=1