Isotropic random flights: random numbers of flights

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Abstract
Isotropic random flights, where the number of individual flights N is random, are studied. N is taken to be governed by a Poisson distribution and also by a negative binomial distribution, each with mean (N). The probability density function of the length of the vector sum is shown to be mixed, in that it contains impulse components (Dirac delta functions) as well as the absolutely continuous component. The limiting density functions are also obtained, and in the negative binomial case lead to the random flight version of the K-density function introduced by Jakeman and collaborators (1976, 1978). Finally, the moments about the origin are explicitly evaluated for both fixed N and random N.