Mathematical Methods for Physics and Engineering - Matematica.NET

PROBABILITYx f(x) (binomial) f(x) (Gaussian)0 0.0001 0.00011 0.0016 0.00142 0.0106 0.00923 0.0425 0.03954 0.1115 0.11195 0.2007 0.20916 0.2508 0.25757 0.2150 0.20918 0.1209 0.11199 0.0403 0.039510 0.0060 0.0092Table 30.4 Comparison of the binomial distribution for n =10andp =0.6with its Gaussian approximation.to obtainf(x) ≈ √ 1 ( x) −x−1/2 ( n − x2πn nn= √ 1 [exp − ( )x + 1 x2πn2 ln) −n+x−1/2p x (1 − p) n−xn − ( n − x + 1 2]+ x ln p +(n − x)ln(1− p) .)lnn − xnBy expanding the argument of the exponential in terms of y = x − np, where1 ≪ y ≪ np and keeping only the dominant terms, it can be shown thatf(x) ≈ √ 1[1√ exp − 1 (x − np) 2 ],2πn p(1 − p) 2 np(1 − p)which is of Gaussian form with µ = np and σ = √ np(1 − p).Thus we see that the value of the Gaussian probability density function f(x) isa good approximation to the probability of obtaining x successes in n trials. Thisapproximation is actually very good even for relatively small n. For example, ifn =10andp =0.6 then the Gaussian approximation to the binomial distributionis (30.105) with µ =10× 0.6 =6andσ = √ 10 × 0.6(1 − 0.6) = 1.549. Theprobability functions f(x) for the binomial and associated Gaussian distributionsfor these parameters are given in table 30.4, and it can be seen that the Gaussianapproximation is a good one.Strictly speaking, however, since the Gaussian distribution is continuous andthe binomial distribution is discrete, we should use the integral of f(x) fortheGaussian distribution in the calculation of approximate binomial probabilities.More specifically, we should apply a continuity correction so that the discreteinteger x in the binomial distribution becomes the interval [x − 0.5, x+0.5] in1186