ABSTRACT
For a random variable X , we have demonstrated how important its first moment E(X) and its second moment E(X2) are. For other values of n also, E(Xn) is a valuable measure both in theory and practice. For example, letting μ = E(X), μ(r)X = E
[ (X−μ)r], we have
that the quantity μ(3)X /σ 3 X is a measure of symmetry of the distribution of X . It is called
the measure of skewness and is zero if the distribution of X is symmetric, negative if it is skewed to the left, and positive if it is skewed to the right (for examples of distributions that are, say, skewed to the right, see Figure 7.12). As another example, μ(4)X /σ
4 X , the measure
of kurtosis, indicates relative flatness of the distribution function of X . For a standard normal distribution function this quantity is 3. Therefore, if μ(4)X /σ
4 X > 3, the distribution
function of X is more peaked than that of standard normal, and if μ(3)X /σ 3 X < 3, it is less
peaked (flatter) than that of the standard normal. The moments of a random variable X give information of other sorts, too. For example, it can be proven that E
(|X |k) < ∞ implies that limn→∞ nkP
(|X | > n) → 0, which shows that if E(|X |k) < ∞, then P(|X | > n) approaches 0 faster than 1/nk as n →∞.
