Motivated by recent results by Nadarajah, Srivastava and Gupta in which the authors introduced a new formula for CDF F_K of the McKay Bessel function distribution containing the modified Bessel function we aim to present new formulae for F_K, especially for certain values of parameters for which their formula fails to be calculated by in-built numerical routines. Also, several computational series formulae are obtained for the CDF of McKay Bessel distribution which contains modified Bessel function one is given in terms of the Horn confluent hypergeometric function and the others by Kampé de Fériet function Marcum Q-function and modified Bessel functions Horn confluent hypergeometric function and finally in terms of incomplete Gamma function γ and Exton’s double hypergeometric X–function.