Mathematical Methods for Physics and Engineering - Matematica.NET

PROBABILITYwhere M Xi (t) istheMGFoff i (x). Now( ) tM Xi =1+ t n n E[X i]+ 1 t 22n 2 E[X2 i ]+···t=1+µ in + 1 2 (σ2 i + µ 2 i ) t2n 2 + ··· ,and as n becomes large( ) ( t µi tM Xi ≈ expnn + 1 t 2 )2 σ2 in 2 ,as may be verified by expanding the exponential up to terms including (t/n) 2 .Thereforen∏(µi tM Z (t) ≈ expn + 1 t 2 ) (∑2 σ2 iin 2 =expµ ∑ )in t + 1 i σ2 i2n 2 t 2 .i=1Comparing this with the form of the MGF for a Gaussian distribution, (30.114),we can see that the probability density function g(z) ofZ tends to a Gaussian distributionwith mean ∑ i µ i/n and variance ∑ i σ2 i /n2 . In particular, if we considerZ to be the mean of n independent measurements of the same random variable X(so that X i = X for i =1, 2,...,n) then, as n →∞, Z has a Gaussian distributionwith mean µ and variance σ 2 /n.We may use the central limit theorem to derive an analogous result to (iii)above for the product W = X 1 X 2 ···X n of the n independent random variablesX i . Provided the X i only take values between zero and infinity, we may writeln W =lnX 1 +lnX 2 + ···+lnX n ,which is simply the sum of n new random variables ln X i . Thus, provided thesenew variables each possess a formal mean and variance, the PDF of ln W willtend to a Gaussian in the limit n →∞, and so the product W will be describedby a log-normal distribution (see subsection 30.9.2).30.11 Joint distributionsAs mentioned briefly in subsection 30.4.3, it is common in the physical sciences toconsider simultaneously two or more random variables that are not independent,in general, and are thus described by joint probability density functions. We willreturn to the subject of the interdependence of random variables after firstpresenting some of the general ways of characterising joint distributions. Wewill concentrate mainly on bivariate distributions, i.e. distributions of only tworandom variables, though the results may be extended readily to multivariatedistributions. The subject of multivariate distributions is large and a detailedstudy is beyond the scope of this book; the interested reader should therefore1196