Mathematical Methods for Physics and Engineering - Matematica.NET

PROBABILITYi =1, 2,...,N, is distributed as X i ∼ Bin(n i ,p)thenZ = X 1 + X 2 + ···+ X N isdistributed as Z ∼ Bin(n 1 + n 2 + ···+ n N ,p), as would be expected since the resultof ∑ i n i trials cannot depend on how they are split up. A similar proof is alsopossible using either the probability or cumulant generating functions.Unfortunately, no equivalent simple result exists for the probability distributionof the difference Z = X − Y of two binomially distributed variables.30.8.2 The geometric and negative binomial distributionsA special case of the binomial distribution occurs when instead of the number ofsuccesses we consider the discrete random variableX = number of trials required to obtain the first success.The probability that x trials are required in order to obtain the first success, issimply the probability of obtaining x − 1 failures followed by one success. If theprobability of a success on each trial is p, thenforx>0f(x) =Pr(X = x) =(1− p) x−1 p = q x−1 p,where q =1− p. This distribution is sometimes called the geometric distribution.The probability generating function for this distribution is given in (30.78). Byreplacing t by e t in (30.78) we immediately obtain the MGF of the geometricdistributionM(t) =pet1 − qe t ,from which its mean and variance are found to beE[X] = 1 p , V[X] = q p 2 .Another distribution closely related to the binomial is the negative binomialdistribution. This describes the probability distribution of the random variableX = number of failures before the rth success.One way of obtaining x failures before the rth success is to have r − 1 successesfollowed by x failures followed by the rth success, for which the probability ispp ···p} {{ }r − 1times× qq ···q × p = p r q x .} {{ }x timesHowever, the first r + x − 1 factors constitute just one permutation of r − 1successes and x failures. The total number of permutations of these r + x − 1objects, of which r − 1 are identical and of type 1 and x are identical and of type1172