Observations on the McKay I Bessel distribution II

Motivated by a wide spectrum of possible applications of the McKay I ν Bessel distribution we aim to present two new formulae for the appropriate distribution function, using the mean–value theorems for integrals: one of Bonnet type and another relying on the stronger version of the Okamura's variant of the second integral mean–value theorem. In both of those results, a point, characteristic for the mean–value theorems, is explicitly presented in terms of the Lambert W function. In addition, the computational efficiency of the newly derived formulae versus initial definition of the mentioned cumulative distribution function is established and the count data problem is resolved for this probability law.