Course in Probability Theory

3.2 PROPERTIES OF MATHEMATICAL EXPECTATION 1 49Another important application is as follows : let µ2 be as in Theorem 3 .2 .3and take f (x, y) to be x + y there . We obtain(16) i(X + Y) = f f (x + Y)Ii2(dx, dy)= ff xµ2(dx, dy) + ff yµ2(dx, dy)o~? zJR~On the other hand, if we take f (x, y) to be x or y, respectively, we obtainand consequently(` (X) = JfxIL2(dx, dy), (`` (Y) = ff yh .2(dx, dy)R2 9?2(17) (-r,(X + Y) _ (-"(X) + e(Y) .This result is a case of the linearity of oc' given but not proved here ; the proofabove reduces this property in the general case of (0, 97, ST) to the correspondingone in the special case (R2, ?Z2, µ2) . Such a reduction is frequentlyuseful when there are technical difficulties in the abstract treatment .We end this section with a discussion of "moments" .Let a be real, r positive, then cf(IX - air) is called the absolute momentof X of order r, about a . It may be +oo ; otherwise, and if r is an integer,~`((X - a)r) is the corresponding moment . If µ and F are, respectively, thep .m . and d .f. of X, then we have by Theorem 3 .2 .2 :('(IX - air) = f Ix - al rµ(dx) = f Ix - air dF(x),00cc `((X . - a)r) = (x - a)rµ(dx) = f (x - a)r dF(x)fJi .00For r = 1, a = 0, this reduces to (`(X), which is also called the mean of X .The moments about the mean are called central moments . That of order 2 isparticularly important and is called the variance, var (X) ; its positive squareroot the standard deviation . 6(X) :var (X) = Q2 (X ) = c`'{ (X - `(X))2) = (,`(X2) { (X ))2 .We note the inequality a2(X) < c`(X2), which will be used a good dealin Chapter 5 . For any positive number p, X is said to belong to LP =LP(Q, :~ , :J/') iff c (jXjP) < oo .