On an additive problem of number theory with random number of summands

Asymptotic formulas for the mean and variance of the number of nonnegative integer solutions of the equation $x_1^m+\dots+x_s^m=N$ are obtained; here $m,s,N$ are integer positive numbers, $m=\mathrm{const}$, and $s$ is a random variable. Cases of the binomial and Poisson distributions of $s-1$ are considered. Proofs are based on the saddle point method. Analogous results for the number of nonnegative integer solutions of the inequality $x_1^m+\dots+x_s^m\le N$ are obtained also.