The task of establishing the law of distribution of a function of random variables according to a given law of distribution of arguments is the main one. The general scheme of reasoning here is as follows. Let be the distribution law. Then we obviously have where is the complete inverse image of the half-interval, i.e. the set of those values of the vector £ from the ZG for which. The last probability can be easily found, since the law of distribution of random variables £ is known. Similarly, in principle, the law of distribution of the vector function of random arguments can be found. The complexity of the implementation of the circuit depends only on the specific type of function (p and the distribution law of the arguments. This chapter is devoted to the implementation of the circuit in specific situations that are important for applications. §1. Functions of one variable Let £ be a random variable, the distribution law of which is given by the distribution function F( (x), rj = If F4(y) is the distribution function of the random variable rj, then the above considerations give FUNCTIONS OF RANDOM VARIABLES where y) denotes the complete inverse image of the half-line (-oo, y).Relation (I) is an obvious consequence of ( *) and for the case under consideration is illustrated in Fig. 1. Monotonic transformation of a random variable Let (p(t) be a continuous monotonic function (for definiteness, monotonically non-increasing) and r) = - For the distribution function Fn(y) we obtain (here is the function , the inverse to the existence of which is ensured by monotonicity and continuity. For monotonically non-decreasing) similar calculations give In particular, if - is linear, then for a > O (Fig. 2) Linear transformations do not change the nature of the distribution, but only affect its parameters. Linear transformation of a random variable uniform on [a, b] Let Linear transformation of a normal random variable Let and in general if Let, for example, 0. From (4) we conclude that Put in the last integral This replacement gives an important identity, which is the source of many interesting applications , can be obtained from relation (3) with Lemma. If is a random variable with a continuous distribution function F^(x), then the random variable r) = is uniform on . We have - monotonically does not decrease and is contained within the limits o Therefore, FUNCTIONS OF RANDOM VARIABLES On the interval we obtain One of the possible ways of using the proven lemma is, for example, the procedure for modeling a random variable with an arbitrary distribution law F((x). As follows from the lemma, for this it is enough to be able to obtain values of uniform on )
