Mathematical Methods for Physics and Engineering - Matematica.NET

30.9 IMPORTANT CONTINUOUS DISTRIBUTIONSNow using Φ(−z) =1− Φ(z) gives( ) µ − 140Φ=1− 0.030 = 0.970.σUsing table 30.3 again, we findµ − 140=1.88. (30.113)σSolving the simultaneous equations (30.112) and (30.113) gives µ = 173.5, σ =17.8. ◭The moment generating function for the Gaussian distributionUsing the definition of the MGF (30.85),M X (t) =E [ e tX] ∫ ∞[]1=−∞ σ √ 2π exp (x − µ)2tx −2σ 2 dx= c exp ( µt + 1 2 σ2 t 2) ,where the final equality is established by completing the square in the argumentof the exponential and writing∫ ∞{1c =−∞ σ √ 2π exp − [x − (µ + σ2 t)] 2 }2σ 2 dx.However, the final integral is simply the normalisation integral for the Gaussiandistribution, and so c = 1 and the MGF is given byM X (t) =exp ( µt + 1 2 σ2 t 2) . (30.114)We showed in subsection 30.7.2 that this MGF leads to E[X] =µ and V [X] =σ 2 ,as required.Gaussian approximation to the binomial distributionWe may consider the Gaussian distribution as the limit of the binomial distributionwhen the number of trials n →∞but the probability of a success p remainsfinite, so that np →∞also. (This contrasts with the Poisson distribution, whichcorresponds to the limit n →∞and p → 0 with np = λ remaining finite.) Inother words, a Gaussian distribution results when an experiment with a finiteprobability of success is repeated a large number of times. We now show howthis Gaussian limit arises.The binomial probability function gives the probability of x successes in n trialsasn!f(x) =x!(n − x)! px (1 − p) n−x .Taking the limit as n →∞(and x →∞) we may approximate the factorials byStirling’s approximationn! ∼ √ ( n) n2πne1185