The main aim of this paper is to present some results related to asymptotic behavior of distribution functions of random variables of chi-square type $X^2_N={Sigma}^N_{i=1};X^2_i$ with degrees of freedom N, where N is a positive integer-valued random variable independent on all standard normally distributed random variables $X_i$. Two ways for computing the distribution functions of chi-square type random variables with random degrees of freedom are considered. Moreover, some tables concerning considered distribution functions are demonstrated in Appendix.