The Moments of a Random Variable
THE MOMENTS OF A RANDOM VARIABLE Definition: Let X be a rv with the range space Rx and let c be any known constant. Then the kth moment of X about the constant c is defined as Mk (X) = E[ (X c)k ]. (12) In the field of statistics only 2 values of c are of interest: c = 0 and c = . Moments about c = 0 are called origin moments and are denoted by k, i.e., k = E(Xk ), where c = 0 has been inserted into equation (12). Moments about the population mean, , are called central moments and are denoted by k, i.e, k = E[ (X )k ], where c = has been inserted into (12).
STATISTICAL INTERPRETATION OF MOMENTS By definition of the kth origin moment, we have: k =
(1) Whether X is discrete or continuous, 1 = E(X) = , i.e., the 1st origin moment is simply the population mean (i.e., 1 measures central tendency). (2) Since the population variance, 2, is the weighted average of deviations from the mean squared over all elements of Rx, then 2 = E[(X )2] = 2. Therefore, the 2nd central moment, 2 = 2, is a measure of dispersion (or variation, or spread) of the population. Further, the 2nd central moment can be expressed in terms of origin moments using the binomial expansion of (X )2, as shown below. 2 = E[ (X )2] = E[(X2 2 X + 2 )] = E(X2) 2 E(X) + 2 = E(X2) 2 = ()2 = 2 . (13) Example 24 (continued). For the exponential density, f(x) = e x, = = 2/2 and = = 1/ so that equation (13) yields 2 = V(x) = 2 = 1/2 . (Note that the exponential pdf is the only Pearsonian statistical model with CVx = 100%.) (3) The 3rd central moment, 3, is a measure of skewness (bear in mind that 3 0 for all symmetrical distributions). If X is continuous, then 3 = E[(X )3] = = 3 + 2 3 (14) For the exponential pdf , we have shown that = 1 = 1/, = 2!/ 2 and you may verify that 3 = 3! /3 = 6 /3.
STATISTICAL INTERPRETATION OF MOMENTS By definition of the kth origin moment, we have: k =
(1) Whether X is discrete or continuous, 1 = E(X) = , i.e., the 1st origin moment is simply the population mean (i.e., 1 measures central tendency). (2) Since the population variance, 2, is the weighted average of deviations from the mean squared over all elements of Rx, then 2 = E[(X )2] = 2. Therefore, the 2nd central moment, 2 = 2, is a measure of dispersion (or variation, or spread) of the population. Further, the 2nd central moment can be expressed in terms of origin moments using the binomial expansion of (X )2, as shown below. 2 = E[ (X )2] = E[(X2 2 X + 2 )] = E(X2) 2 E(X) + 2 = E(X2) 2 = ()2 = 2 . (13) Example 24 (continued). For the exponential density, f(x) = e x, = = 2/2 and = = 1/ so that equation (13) yields 2 = V(x) = 2 = 1/2 . (Note that the exponential pdf is the only Pearsonian statistical model with CVx = 100%.) (3) The 3rd central moment, 3, is a measure of skewness (bear in mind that 3 0 for all symmetrical distributions). If X is continuous, then 3 = E[(X )3] = = 3 + 2 3 (14) For the exponential pdf , we have shown that = 1 = 1/, = 2!/ 2 and you may verify that 3 = 3! /3 = 6 /3.