On Some Bivariate Probability Models Applicable to Traffic Accidents and Fatalities

Abstract

Suppose X denotes the number of injury accidents on a given stretch of highway in a given period, Y denotes the number of these accidents which resulted in one or more fatalities and Z denotes the number of fatalities recorded among the X accidents. In this Review, Leiter amd Hamdan (1973) suggested that the joint distribution of the number of accidents and the number of fatal accidents can be expressed by a Poisson-Bernoulli model (Poisson-Binomial in their terminology) and the joint distribution of the number of accidents and the number of fatalities by a Poisson-Poisson, P-P, distribution. In the present paper an alternative model, the Poisson-Binomial, P-B, model, is proposed to express the relationship between the number of accidents and the number of fatalities. Various properties of this model were investigated including the derivation of the conditional probability generating function (p.g.f.) using the basic relation between the Bell polynomials and the derivatives of a composite function (see for example Riordan, 1958, p. 34). For comparison purposes the P-B model is fitted to the same set of accident data used by Leiter and Hamdan. The present representation of Z, the number of fatalities, as a sum of X binomial variables, each with parameters n and p, means that the n passengers in a motor vehicle involved in an injury accident play the role of n independent Bernoulli trials; this assumption of independence is, of course, rather unrealistic since the fate (fatality or not) of a passenger is correlated with that of his co-passengers. Nevertheless the Poisson-Binomial model is almost as good as the Poisson-Poisson model, even for small n = 2, 3. As n gets larger (n = 4 or 5) the P-B model approaches the P-P model, as reflected in the values of the X2-criterion. This is expected since the values ofp, the probability of a passenger being killed in an injury accident, are small and so the binomial distribution can be approximated by the Poisson distribution, even for moderate values of n. The P-P model was further studied and expressions for the bivariate p.g.f. and the conditional p.g.f.'s were obtained. Finally, it is pointed out that the Poisson-Bernoulli model is a special case of the well known bivariate Poisson distribution (see for example Johnson and Kotz, 1968, p. 297 and the references therein).