Analysis of the spectrum of random symmetric Boolean functions

General probabilistic model of a random symmetric Boolean function of $n$ variables is proposed. The characteristic function of the Walsh spectrum of a random symmetric Boolean function is defined; exact and asymptotic distributions of some spectrum characteristics as $n\to\infty$ are obtained in the case of the parametric measure. The basic properties of the Krawtchouk's polynomials (which are used in proofs) are reviewed.