Mathematical Methods for Physics and Engineering - Matematica.NET

30.7 GENERATING FUNCTIONSThe MGF will exist for all values of t provided that X is bounded and alwaysexists at the point t =0whereM(0) = E(1) = 1.It will be apparent that the PGF and the MGF for a random variable Xare closely related. The former is the expectation of t X whilst the latter is theexpectation of e tX :Φ X (t) =E [ t X] , M X (t) =E [ e tX] .The MGF can thus be obtained from the PGF by replacing t by e t , and viceversa. The MGF has more general applicability, however, since it can be usedwith both continuous and discrete distributions whilst the PGF is restricted tonon-negative integer distributions.As its name suggests, the MGF is particularly useful for obtaining the momentsof a distribution, as is easily seen by noting thatE [ e tX] = E[1+tX + t2 X 2 ]+ ···2!=1+E[X]t + E [ X 2] t 22! + ··· .Assuming that the MGF exists for all t around the point t = 0, we can deducethat the moments of a distribution are given in terms of its MGF by∣E[X n ]= dn M X (t) ∣∣∣t=0dt n . (30.86)Similarly, by substitution in (30.51), the variance of the distribution is given byV [X] =M ′′ X(0) − [ M ′ X(0) ] 2, (30.87)where the prime denotes differentiation with respect to t.◮The MGF for the Gaussian distribution (see the end of subsection 30.9.1) is given byM X (t) =exp ( µt + 1 2 σ2 t 2) .Find the expectation and variance of this distribution.Using (30.86),M X(t) ′ = ( µ + σ 2 t ) exp ( µt + 1 2 σ2 t 2) ⇒ E[X] =M X(0) ′ = µ,M X(t) ′′ = [ σ 2 +(µ + σ 2 t) 2] exp ( µt + 1 2 σ2 t 2) ⇒ M X(0) ′′ = σ 2 + µ 2 .Thus, using (30.87),V [X] =σ 2 + µ 2 − µ 2 = σ 2 .That the mean is found to be µ and the variance σ 2 justifies the use of these symbols inthe Gaussian distribution. ◭The moment generating function has several useful properties that follow fromits definition and can be employed in simplifying calculations.1163