Mathematical Methods for Physics and Engineering - Matematica.NET

30.5 PROPERTIES OF DISTRIBUTIONS|x − µ| ≥c. From (30.48), we find that∫∫σ 2 ≥ (x − µ) 2 f(x) dx ≥ c 2|x−µ|≥c|x−µ|≥cf(x) dx. (30.49)The first inequality holds because both (x − µ) 2 and f(x) are non-negative forall x, and the second inequality holds because (x − µ) 2 ≥ c 2 over the range ofintegration. However, the RHS of (30.49) is simply equal to c 2 Pr(|X − µ| ≥c),and thus we obtain the required inequalityPr(|X − µ| ≥c) ≤ σ2c 2 .A similar derivation may be carried through for the case of a discrete randomvariable. Thus, for any distribution f(x) that possesses a variance we have, forexample,Pr(|X − µ| ≥2σ) ≤ 1 4and Pr(|X − µ| ≥3σ) ≤ 1 9 .30.5.4 MomentsThe mean (or expectation) of X is sometimes called the first moment of X, sinceit is defined as the sum or integral of the probability density function multipliedby the first power of x. By a simple extension the kth moment of a distributionis defined by{∑µ k ≡ E[X k j]=xk j f(x j) for a discrete distribution,∫x k f(x) dx for a continuous distribution.(30.50)For notational convenience, we have introduced the symbol µ k to denote E[X k ],the kth moment of the distribution. Clearly, the mean of the distribution is thendenoted by µ 1 , often abbreviated simply to µ, as in the previous subsection, asthis rarely causes confusion.A useful result that relates the second moment, the mean and the variance ofa distribution is proved using the properties of the expectation operator:V [X] =E [ (X − µ) 2]= E [ X 2 − 2µX + µ 2]= E [ X 2] − 2µE[X]+µ 2= E [ X 2] − 2µ 2 + µ 2In alternative notations, this result can be written= E [ X 2] − µ 2 . (30.51)〈(x − µ) 2 〉 = 〈x 2 〉−〈x〉 2 or σ 2 = µ 2 − µ 2 1.1147