12.4. The distribution law of the function of two random variables

The task of determining the distribution law of a function of several random arguments is much more complicated than a similar problem for a function of one argument. Here we present a general method for solving this problem for the simplest case of a function of two arguments.

There is a system of two continuous random variables. with distribution density . Random value associated with and functional dependence:

.

It is required to find the distribution law .

To solve the problem, we use a geometrical interpretation similar to the one we used in the case of a single argument. Function it’s no longer a curve, but a surface (fig. 12.4.1).

Fig. 12.4.1.

Find the distribution function of :

. (12.4.1)

Draw a plane parallel to the plane , on distance from her. This plane will cross the surface. on some curve . Design the curve on the plane . This projection, whose equation will divide the plane into two areas; for one of them is the height of the surface above the plane will be less, and for another - more . Denote the area for which this height is less . To satisfy inequality (12.4.1), a random point obviously should get into the area ; Consequently,

. (12.4.2)

In the expression (12.4.2) value enters implicitly through the limits of integration.

Differentiating by , we obtain the distribution density :

.

Knowing the specific type of function , you can express the limits of integration through and write the expression explicitly.

In order to find the distribution law of the function of two arguments, there is no need to construct a surface each time. , just as it is done in fig. 12.4.1, and cross it by a plane parallel to . In practice, it is enough to build on the plane curve whose equation , be aware of which side of this curve and for what and integrate by region , for which .

Example. System of quantities subject to the distribution law with density . Magnitude there is a product of quantities :

.

Find the density distribution .

Decision. Let's set some value and build on the plane curve whose equation (fig. 12.4.2). This is a hyperbole whose asymptotes coincide with the axes of coordinates. Region in fig. 12.4.2 shaded.

Fig. 12.4.2.

Distribution function of magnitude has the form:

.

Differentiating this expression by , we have:

. (12.4.3)