Poisson Processes and a Bessel Function Integral

The probability of winning a simple game of competing Poisson processes turns out to be equal to the well-known Bessel function integral $J(x,y)$ (cf. Y. L. Luke, Integrals of Bessel Functions, McGraw-Hill, New York, 1962). Several properties of J, some of which seem to be new, follow quite easily from this probabilistic interpretation. The results are applied to the random telegraph process as considered by Kac [Rocky Mountain J. Math., 4 (1974), pp. 497–509].