Mathematical Methods for Physics and Engineering - Matematica.NET

PROBABILITY30.5.3 Variance and standard deviationThe variance of a distribution, V [X], also written σ 2 , is defined byV [X] =E [ {∑(X − µ) 2] j=(x j − µ) 2 f(x j )∫(x − µ) 2 f(x) dxfor a discrete distribution,for a continuous distribution.(30.48)Here µ has been written for the expectation value E[X] ofX. As in the case ofthe mean, unless the series and the integral in (30.48) converge the distributiondoes not have a variance. From the definition (30.48) we may easily derive thefollowing useful properties of V [X]. If a and b are constants then(i) V [a] =0,(ii) V [aX + b] =a 2 V [X].The variance of a distribution is always positive; its positive square root isknown as the standard deviation of the distribution and is often denoted by σ.Roughly speaking, σ measures the spread (about x = µ) of the values that X canassume.◮Find the standard deviation of the PDF for the distance from the origin of the electronwhose wavefunction was discussed in the previous two examples.Inserting the expression (30.47) for the PDF f(r) into (30.48), the variance of the randomvariable R is given by∫ ∞V [R] = (r − µ) 2 4r2 e −2r/a 0dr = 4 (r 4 − 2r 3 µ + r 2 µ 2 )e −2r/a 0dr,0 a 3 0a 3 0 0where the mean µ = E[R] =3a 0 /2. Integrating each term in the integrand by parts weobtain∫ ∞V [R] =3a 2 0 − 3µa 0 + µ 2 = 3a2 04 .Thus the standard deviation of the distribution is σ = √ 3a 0 /2. ◭We may also use the definition (30.48) to derive the Bienaymé–Chebyshevinequality, which provides a useful upper limit on the probability that randomvariable X takes values outside a given range centred on the mean. Let us considerthe case of a continuous random variable, for which∫Pr(|X − µ| ≥c) = f(x) dx,|x−µ|≥cwhere the integral on the RHS extends over all values of x satisfying the inequality1146