Mathematical Expectation and Moments If g(X) is a function of a RV X, then the expected value of g(X) is defined by The moments about the mean value u = E(X) i. e., if X is discrete RV a = H = E(X) Fig(X)} =1 6 are called central moments and denoted by up. g(x) . f(x) dx, if X is continuous RV Thus, up = E((X -u)"), where u = E(X) Note : Ela . g(X) + b . h(X)] = a . Elg(X)} + b . Eth(X)} and E(c) = c if X is discrete RV Here a, b and c are constants. Hr =1 ((x - u) . f(x) dx, if X is continuous RV In particular, E(ax + b) = a . E(X) + b. - Do Moments We note that when r = 1, then #1 = E(X - H) = E(X) - 1 = H -# = 0 The rth order non - central moments about the value 'a' of a RV X, denoted The rth order non - central, simple and central absolute moments are defined by u, is defined by by E{|X - al"), E{IX|"} and E( IX - ul"} respectively. M, = E{(X - a)") [ (x - a) . Pi if X is discrete RV H, = (x - a)" . f(x) dx, if X is continuous RV When a = 0, then u, = E(X") is called the rth order simple moment. We note that when a = 0 and r = 1, then uj = E(X) is the mean of RV X. Simple moments are also called raw moments or ordinary moments