Abstract: In this paper, a discrete system described by a one-dimensional map is constructed in terms of our thought on the mechanism of chaos. Theory and experiment (including the related function analysis, spectrum analysis, statistical test) both confirm that when λ=√3, the system is of infinite-periodicity, non-asymptoticity, and chaoticity. According to our opinion, a chaotic solution possesses its definite distribution, although it is very sensitive to its initial condition. In this paper our system is shown to have a uniform distribution. Thus we use it to design a random number generator. The numbers generated have good uniformity and independence. Moreover, we introduce the method of inverse transformation which makes the generation of random signals of strictly monotone and continuous distributions solved completely. The designs for the generators of pseudo-random signals with uniform distribution on[a, b], exponetial distribution e (λ, μ), β-distribution β(p, q) and Gauss distribution N(μ, b) are given. Statistical tests show our new design method is both pracficable and economical.