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

Lecture



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.   12.4.  The distribution law of the function of two random variables with distribution density   12.4.  The distribution law of the function of two random variables . Random value   12.4.  The distribution law of the function of two random variables associated with   12.4.  The distribution law of the function of two random variables and   12.4.  The distribution law of the function of two random variables functional dependence:

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

It is required to find the distribution law   12.4.  The distribution law of the function of two random variables .

To solve the problem, we use a geometrical interpretation similar to the one we used in the case of a single argument. Function   12.4.  The distribution law of the function of two random variables it’s no longer a curve, but a surface (fig. 12.4.1).

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

Fig. 12.4.1.

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

  12.4.  The distribution law of the function of two random variables . (12.4.1)

Draw a plane   12.4.  The distribution law of the function of two random variables parallel to the plane   12.4.  The distribution law of the function of two random variables , on distance   12.4.  The distribution law of the function of two random variables from her. This plane will cross the surface.   12.4.  The distribution law of the function of two random variables on some curve   12.4.  The distribution law of the function of two random variables . Design the curve   12.4.  The distribution law of the function of two random variables on the plane   12.4.  The distribution law of the function of two random variables . This projection, whose equation   12.4.  The distribution law of the function of two random variables will divide the plane   12.4.  The distribution law of the function of two random variables into two areas; for one of them is the height of the surface above the plane   12.4.  The distribution law of the function of two random variables will be less, and for another - more   12.4.  The distribution law of the function of two random variables . Denote   12.4.  The distribution law of the function of two random variables the area for which this height is less   12.4.  The distribution law of the function of two random variables . To satisfy inequality (12.4.1), a random point   12.4.  The distribution law of the function of two random variables obviously should get into the area   12.4.  The distribution law of the function of two random variables ; Consequently,

  12.4.  The distribution law of the function of two random variables . (12.4.2)

In the expression (12.4.2) value   12.4.  The distribution law of the function of two random variables enters implicitly through the limits of integration.

Differentiating   12.4.  The distribution law of the function of two random variables by   12.4.  The distribution law of the function of two random variables , we obtain the distribution density   12.4.  The distribution law of the function of two random variables :

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

Knowing the specific type of function   12.4.  The distribution law of the function of two random variables , you can express the limits of integration through   12.4.  The distribution law of the function of two random variables and write the expression   12.4.  The distribution law of the function of two random variables explicitly.

In order to find the distribution law of the function of two arguments, there is no need to construct a surface each time.   12.4.  The distribution law of the function of two random variables , just as it is done in fig. 12.4.1, and cross it by a plane parallel to   12.4.  The distribution law of the function of two random variables . In practice, it is enough to build on the plane   12.4.  The distribution law of the function of two random variables curve whose equation   12.4.  The distribution law of the function of two random variables , be aware of which side of this curve   12.4.  The distribution law of the function of two random variables and for what   12.4.  The distribution law of the function of two random variables and integrate by region   12.4.  The distribution law of the function of two random variables , for which   12.4.  The distribution law of the function of two random variables .

Example. System of quantities   12.4.  The distribution law of the function of two random variables subject to the distribution law with density   12.4.  The distribution law of the function of two random variables . Magnitude   12.4.  The distribution law of the function of two random variables there is a product of quantities   12.4.  The distribution law of the function of two random variables :

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

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

Decision. Let's set some value   12.4.  The distribution law of the function of two random variables and build on the plane   12.4.  The distribution law of the function of two random variables curve whose equation   12.4.  The distribution law of the function of two random variables (fig. 12.4.2). This is a hyperbole whose asymptotes coincide with the axes of coordinates. Region   12.4.  The distribution law of the function of two random variables in fig. 12.4.2 shaded.

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

Fig. 12.4.2.

Distribution function of magnitude   12.4.  The distribution law of the function of two random variables has the form:

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

Differentiating this expression by   12.4.  The distribution law of the function of two random variables , we have:

  12.4.  The distribution law of the function of two random variables . (12.4.3)


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis