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%.)
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(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. Hence, substitution into (14) yields
3 = + = . In order to show that 3 = 3! /3, it will be easier to use the partial differentiation and the fact that 3 = = .
If X is measured in hours, the units of 3 are expressed in terms of hours3. To obtain a unit-less measure of asymmetry (for comparative purposes), we standardize 3 to obtain the coefficient of skewness (most authors refer to this coefficient simply as skewness) given below:
3 = 3 / 3 .
Furthermore , some authors (such as Maurice Kendall) use the notation 1 = for 3 = 3 / 3 ,i.e., 1 = = 3 = 3 / 3
For example, the value of coefficient of skewness ( or simply skewness) for the exponential pdf is 3 = 3 / 3 = = 2.00, which is unit-less. It is not common for the value of 3 = 3 / 3 to lie outside the interval [2, 2].
Karl Pearson proposed the measure Sk = ( MO)/ or ( x0.50 )/ to evaluate the asymmetry of a statistical distribution. When a distribution is symmetrical, not only 3 = 3 / 3 0, but also the values of , x0.50, and the Modal point all three coincide. When 3 = 3 / 3 > 0, i.e., the distribution is positively skewed, then it turns out that invariably > x0.50 > MO. However, when a statistical distribution is negatively skewed (i.e., 3 < 0), then always < x0.50 < MO. For the exponential pdf, Sk = ( x0.50 )/ = 0.30685282.
Exercise 26(g).
Compute the value of skewness, 3 = 3 / 3 , for the density function of the rv acidity and also the value of Sk.
(4) The 4th central moment is a measure of Kurtosis (peaked-ness in the middle and heavy prs at the tails) and is given by (for a continuous rv)
4 = =
This last expression after simplifying reduces to
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4 = 4 + 6 2 3 4 (15)
For the exponential density function = =
= = = 4 !/ 4 = 24/ 4 . Recall that for the exponential density = 3 !/ 3 , = 2 !/ 2 and = 1/. Substituting these into equation (15), for the exponential density, we obtain 4 = 9/ 4 .
Again to obtain a unit-less measure of kurtosis we standardize 4 to obtain a (coefficient) of kurtosis defined as
4 = 2 = 4 / 4 .
For the exponential density function, the above measure of kurtosis is given by 4 = 9.00. Most statistical distributions have their 4 values within the interval [1.1, 9]. However, most statistical software normalize the value of 4 = 2 = 4 / 4 by 3 and use the terminology kurtosis = (4 / 4) 3 so that the exponential kurtosis is equal to 6.
Exercise 26 (h).
Compute the value of Kurtosis for the rv Acidity, which was assumed to possess a triangular pdf. ANSWER: All Triangular distributions in the universe have a Kurtosis 4 3 = 4 / 4 3= 0.60000. (i) For the rv Acidity, obtain the value of the REL-IQR = IQR/ .
(5) Recall from chapter 1 that the IQR = Q3 Q1 = x0.75 x0.25 is another measure of variability. To obtain the 25th percentile (or the 0.25 quantile) of the exponential density, we set its cdf equal to 0.25 and solve for the corresponding value of x : 1 e Q1 = 0.25 Q1 = 0.2876821/ , and similarly Q3 = 1.38629436 / . Hence, the value of the IQR for the exponential density is IQR = 1.0986123/. Therefore, all exponential density functions in the universe have a relative IQR equal to IQR/ = REL-IQR = 1.0986123.
