Nontransitive triplets of continuous random variables and their applications

The phenomenon of nontransitivity of the stochastic precedence relation for three independent random variables with distributions from some classes of continuous distributions is studied. Initially, this question was posed in connection with the application in strength theory. With paired comparisons of iron bars from three factories, a paradoxical situation may arise when the bars from the first factory are “worse” than the bars from the second factory, the bars from the second factory are “worse” than the bars from the third factory, and the bars from the third factory are “worse” than the bars from the first factory. Further, the nontransitivity topic gained popularity for the example of the so-called nontransitive dice; however, this led to its narrowing down to discrete random variables with finite sets of values. The paper presents that for mixtures of normal and exponential distributions, nontransitivity is possible in a wide range of parameters. Specific features of the mutual arrangement of the graphs of the distribution functions in these cases are indicated.