Stat 5101 Lecture Notes - School of Statistics

2.4. MOMENTS 39holds for all real-valued functions g such that the expectations exist and if bothexpectations exist or neither. In this case we will say that X and Y are equalin distribution and use the notationX D = Y.This notation is a bit misleading, since it actually says nothing about X andY themselves, but only about their distributions. What is does imply is any ofthe followingP X = P YF X = F Yf X = f Ythat is, X and Y have the same probability measure, the same distributionfunction, or the same probability density. What it does not imply is anythingabout the values of X and Y themselves, which like all random variables arefunctions on the sample space. It may be that X(ω) is not equal to Y (ω) forany ω. Nevertheless, the notation is useful.We say a real-valued random variable X is symmetric about zero if X and−X have the same distribution, that is, ifX D = −X.Note that this is an example of the variables themselves not being equal. Clearly,X(ω) ≠ −X(ω) unless X(ω) = 0, which may occur with probability zero (willoccur with probability zero whenever X is a continuous random variable).We say a real-valued random variable X is symmetric about a point a if X −ais symmetric about zero, that is, ifX − a D = a − X.The point a is called the center of symmetry of X. (Note: Lindgren, definitionon p. 94, gives what is at first glance a completely unrelated definition of thisconcept. The two definitions, his and ours, do in fact define the same concept.See Problem 2-11.)Some of the most interesting probability models we will meet later involvesymmetric random variables, hence the following theorem is very useful.Theorem 2.10. Suppose a real-valued random variable X is symmetric aboutthe point a. If the mean of X exists, it is equal to a. Every higher odd integercentral moment of X that exists is zero.In notation, the two assertions of the theorem areE(X) =µ=aandµ 2k−1 = E{(X − µ) 2k−1 } =0, for any positive integer k.The proof is left as an exercise (Problem 2-10).