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8.3. Distribution density of a system of two random variables

Lecture



Introduced in the previous   8.3.  Distribution density of a system of two random variables system characteristic - the distribution function - exists for systems of any random variables, both discontinuous and continuous. The main practical importance are systems of continuous random variables. The distribution of a system of continuous quantities is usually characterized not by a distribution function, but by a distribution density.

Introducing the distribution density for one random variable, we defined it as the limit of the ratio of the probability of hitting a small area to the length of this area with its unlimited decrease. Similarly, we determine the distribution density of a system of two quantities.

Let there be a system of two continuous random variables.   8.3.  Distribution density of a system of two random variables which is interpreted by a random point on the plane   8.3.  Distribution density of a system of two random variables . Consider a small rectangle on this plane.   8.3.  Distribution density of a system of two random variables with the parties   8.3.  Distribution density of a system of two random variables and   8.3.  Distribution density of a system of two random variables adjacent to the point with coordinates   8.3.  Distribution density of a system of two random variables (fig. 8.3.1). The probability of hitting this rectangle by the formula (8.2.2) is equal to

  8.3.  Distribution density of a system of two random variables

  8.3.  Distribution density of a system of two random variables

Fig. 8.3.1

Divide the probability of hitting the rectangle.   8.3.  Distribution density of a system of two random variables on the area of ​​this rectangle and go to the limit at   8.3.  Distribution density of a system of two random variables and   8.3.  Distribution density of a system of two random variables :

  8.3.  Distribution density of a system of two random variables (8.3.1)

Suppose the function   8.3.  Distribution density of a system of two random variables not only continuous, but differentiable; then the right side of the formula (8.3.1) is the second mixed partial derivative of the function   8.3.  Distribution density of a system of two random variables by   8.3.  Distribution density of a system of two random variables and   8.3.  Distribution density of a system of two random variables . We denote this derivative   8.3.  Distribution density of a system of two random variables :

  8.3.  Distribution density of a system of two random variables (8.3.2)

Function   8.3.  Distribution density of a system of two random variables called the distribution density of the system.

Thus, the distribution density of the system is the limit of the ratio of the probability of hitting a small rectangle to the area of ​​this rectangle, when both of its dimensions tend to zero; it can be expressed as the second mixed partial derivative of the distribution function of the system with respect to both arguments.

If we use the "mechanical" interpretation of the distribution of the system as the distribution of a unit mass over the plane   8.3.  Distribution density of a system of two random variables function   8.3.  Distribution density of a system of two random variables is the mass density distribution at   8.3.  Distribution density of a system of two random variables .

  8.3.  Distribution density of a system of two random variables

Fig. 8.3.2

Geometrically function   8.3.  Distribution density of a system of two random variables can be depicted as a surface (Fig. 8.3.2). This surface is similar to the distribution curve for one random variable and is called the distribution surface.

  8.3.  Distribution density of a system of two random variables

Fig. 8.3.3

If you cross the distribution surface   8.3.  Distribution density of a system of two random variables plane parallel to the plane   8.3.  Distribution density of a system of two random variables , and project the resulting section on the plane   8.3.  Distribution density of a system of two random variables , we get a curve, at each point of which the distribution density is constant. Such curves are called equal density curves. Curves of equal density, obviously, represent the horizontal surface distribution. It is often convenient to set the distribution of a family of curves of equal density.

Considering the density of distribution   8.3.  Distribution density of a system of two random variables for one random variable, we introduced the concept of "probability element"   8.3.  Distribution density of a system of two random variables . This is the probability of hitting a random variable.   8.3.  Distribution density of a system of two random variables on the elementary plot   8.3.  Distribution density of a system of two random variables adjacent to the point   8.3.  Distribution density of a system of two random variables . A similar concept of the “probability element” is introduced for a system of two quantities. The element of probability in this case is the expression

  8.3.  Distribution density of a system of two random variables .

Obviously, the element of probability is nothing but the probability of falling into an elementary rectangle with sides   8.3.  Distribution density of a system of two random variables ,   8.3.  Distribution density of a system of two random variables adjacent to the point   8.3.  Distribution density of a system of two random variables (fig. 8.3.3).

This probability is equal to the volume of the elementary parallelepiped bounded above by the surface   8.3.  Distribution density of a system of two random variables and based on the elementary rectangle   8.3.  Distribution density of a system of two random variables (fig. 8.3.4).

Using the concept of an element of probability, we derive an expression for the probability of a random point hitting an arbitrary region   8.3.  Distribution density of a system of two random variables . This probability can obviously be obtained by summing (integrating) probability elements over the entire region   8.3.  Distribution density of a system of two random variables :

  8.3.  Distribution density of a system of two random variables (8.3.3)

Geometrically probability of hitting the area   8.3.  Distribution density of a system of two random variables depicted by the volume of a cylindrical body   8.3.  Distribution density of a system of two random variables bounded above the surface of the distribution and based on the area   8.3.  Distribution density of a system of two random variables (fig. 8.3.5).

  8.3.  Distribution density of a system of two random variables

Fig. 8.3.4 Figure 8.3.5

The general formula (8.3.3) implies the formula for the probability of hitting the rectangle   8.3.  Distribution density of a system of two random variables limited by abscissas   8.3.  Distribution density of a system of two random variables and   8.3.  Distribution density of a system of two random variables and ordinates   8.3.  Distribution density of a system of two random variables and   8.3.  Distribution density of a system of two random variables (fig. 8.3.5);

  8.3.  Distribution density of a system of two random variables . (8.3.4)

We use the formula (8.3.4) in order to express the distribution function of the system   8.3.  Distribution density of a system of two random variables through density distribution   8.3.  Distribution density of a system of two random variables . Distribution function   8.3.  Distribution density of a system of two random variables there is a chance of falling into an infinite quadrant; the latter can be considered as a rectangle bounded by abscissas -   8.3.  Distribution density of a system of two random variables and   8.3.  Distribution density of a system of two random variables and ordinates -   8.3.  Distribution density of a system of two random variables and   8.3.  Distribution density of a system of two random variables . By the formula (8.3.4) we have:

  8.3.  Distribution density of a system of two random variables . (8.3.5)

It is easy to verify the following properties of the distribution density of the system:

1. The distribution density of a system is a non-negative function:

  8.3.  Distribution density of a system of two random variables .

This is clear from the fact that the distribution density is the limit of the ratio of two non-negative values: the probability of hitting the rectangle and the area of ​​the rectangle — and, therefore, cannot be negative.

2. The double integral in the infinite limits of the distribution density of the system is equal to one:

  8.3.  Distribution density of a system of two random variables (8.3.6)

This is evident from the fact that the integral (8.3.6) is nothing more than the probability of hitting the entire plane.   8.3.  Distribution density of a system of two random variables i.e. probability of a reliable event.

Geometrically, this property means that the total volume of the body bounded by the surface distribution and the plane   8.3.  Distribution density of a system of two random variables , is equal to one.

Example 1. A system of two random variables   8.3.  Distribution density of a system of two random variables subject to the distribution law with density

  8.3.  Distribution density of a system of two random variables .

Find distribution function   8.3.  Distribution density of a system of two random variables . Determine the probability of hitting a random point.   8.3.  Distribution density of a system of two random variables in square   8.3.  Distribution density of a system of two random variables (fig. 8.3.6).

  8.3.  Distribution density of a system of two random variables

Fig. 8.3.6

Decision. Distribution function   8.3.  Distribution density of a system of two random variables we find by the formula (8.3.5).

  8.3.  Distribution density of a system of two random variables .

Probability of hitting a rectangle   8.3.  Distribution density of a system of two random variables we find by the formula (8.3.4):

  8.3.  Distribution density of a system of two random variables .

Example 2. System distribution surface   8.3.  Distribution density of a system of two random variables is a straight circular cone, the base of which is a circle of radius   8.3.  Distribution density of a system of two random variables centered at the origin. Write an expression for the density of distribution. Determine the probability that a random point   8.3.  Distribution density of a system of two random variables will fall into a circle   8.3.  Distribution density of a system of two random variables radius   8.3.  Distribution density of a system of two random variables (fig. 8.3.7), and   8.3.  Distribution density of a system of two random variables .

  8.3.  Distribution density of a system of two random variables

Fig. 8.3.7 Figure 8.3.8

Decision. The expression of the density of the distribution inside the circle   8.3.  Distribution density of a system of two random variables we find from fig. 8.3.8:

  8.3.  Distribution density of a system of two random variables ,

Where   8.3.  Distribution density of a system of two random variables - the height of the cone. Magnitude   8.3.  Distribution density of a system of two random variables determined so that the volume of the cone was equal to one:   8.3.  Distribution density of a system of two random variables from where

  8.3.  Distribution density of a system of two random variables ,

and

  8.3.  Distribution density of a system of two random variables .

Probability of hitting the circle   8.3.  Distribution density of a system of two random variables determined by the formula (8.3.4):

  8.3.  Distribution density of a system of two random variables . (8.3.7)

To calculate the integral (8.3.7), it is convenient to go to the polar coordinate system   8.3.  Distribution density of a system of two random variables :

  8.3.  Distribution density of a system of two random variables .


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis