Nontransitive triplets of continuous random variables

We study the problem of the nontransitivity of the stochastic precedence relation $ P(X <Y)>1/2$ for three independent random variables taken from a given class of continuous distributions. We present some criteria which help us to prove that for many classical distributions the transitivity holds. We give examples of distributions such that for their mixtures the nontransitivity is possible. Various applications are discussed.