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8.2. The distribution function of the system of two random variables

Lecture



The distribution function of a system of two random variables   8.2.  The distribution function of the system of two random variables called the probability of joint fulfillment of two inequalities   8.2.  The distribution function of the system of two random variables and   8.2.  The distribution function of the system of two random variables :

  8.2.  The distribution function of the system of two random variables . (8.2.1)

If we use for the geometric interpretation of the system the image of a random point, then the distribution function   8.2.  The distribution function of the system of two random variables there is nothing like the probability of hitting a random point   8.2.  The distribution function of the system of two random variables in an infinite quadrant with a vertex at a point   8.2.  The distribution function of the system of two random variables lying to the left and below it (Fig. 8.2.1). In a similar interpretation, the distribution function of one random variable   8.2.  The distribution function of the system of two random variables - we denote it   8.2.  The distribution function of the system of two random variables - represents the probability of hitting a random point in the half-plane, bounded to the right by the abscissa   8.2.  The distribution function of the system of two random variables (Fig. 8.2.2); single value distribution function   8.2.  The distribution function of the system of two random variables - probability of hitting the half-plane bounded by ordinate y (Fig. 8.2.3).

  8.2.  The distribution function of the system of two random variables

Fig. 8.2.1

AT   8.2.  The distribution function of the system of two random variables 5.2 we gave the main properties of the distribution function   8.2.  The distribution function of the system of two random variables for one random variable. We formulate similar properties for the distribution function of a system of random variables and again use the geometric interpretation to illustrate these properties.

1. Distribution function   8.2.  The distribution function of the system of two random variables there is a non-decreasing function of both its arguments, i.e.

at   8.2.  The distribution function of the system of two random variables ;

at   8.2.  The distribution function of the system of two random variables .

In this property of the function   8.2.  The distribution function of the system of two random variables can be clearly seen using the geometric interpretation of the distribution function as the probability of falling into a quadrant with vertex   8.2.  The distribution function of the system of two random variables (Fig. 8.2.1). Indeed, increasing   8.2.  The distribution function of the system of two random variables (by shifting the right edge of the quadrant to the right) or by increasing   8.2.  The distribution function of the system of two random variables (by shifting the upper boundary upward), we obviously cannot reduce the probability of falling into this quadrant.

  8.2.  The distribution function of the system of two random variables   8.2.  The distribution function of the system of two random variables

Fig. 8.2.2 Figure 8.2.3

2. Everywhere   8.2.  The distribution function of the system of two random variables the distribution function is zero:

  8.2.  The distribution function of the system of two random variables .

In this property we clearly see, unlimitedly pushing the right edge of the quadrant to the left   8.2.  The distribution function of the system of two random variables or down its upper limit   8.2.  The distribution function of the system of two random variables or doing it simultaneously with both boundaries; while the probability of hitting the quadrant tends to zero.

3. With one of the arguments equal to   8.2.  The distribution function of the system of two random variables , the function of the distributed system becomes the distribution function of a random variable corresponding to another argument:

  8.2.  The distribution function of the system of two random variables ,

Where   8.2.  The distribution function of the system of two random variables - respectively, the distribution function of the random, the distribution function of values   8.2.  The distribution function of the system of two random variables and   8.2.  The distribution function of the system of two random variables .

This property of the distribution function can be clearly seen by shifting one or another of the boundaries of the quadrant by   8.2.  The distribution function of the system of two random variables ; while in the limit the quadrant turns into a half-plane, the probability of falling into which is the distribution function of one of the quantities included in the system.

4. If both arguments are equal   8.2.  The distribution function of the system of two random variables , the distribution function of the system is equal to one:

  8.2.  The distribution function of the system of two random variables .

Indeed, with   8.2.  The distribution function of the system of two random variables ,   8.2.  The distribution function of the system of two random variables quadrant with vertex   8.2.  The distribution function of the system of two random variables in the limit turns into the whole plane   8.2.  The distribution function of the system of two random variables , hit in which there is a reliable event.

When considering the laws of distribution of individual random variables (Chapter 5), we derived an expression for the probability of a random variable falling within a given segment. We expressed this probability both through the distribution function and through the distribution density.

In a similar way for a system of two random variables is the question of the probability of hitting a random point   8.2.  The distribution function of the system of two random variables within the specified area   8.2.  The distribution function of the system of two random variables on surface   8.2.  The distribution function of the system of two random variables (Fig.8.2.4).

  8.2.  The distribution function of the system of two random variables

Fig. 8.2.4

Let's arrange an event consisting in hitting a random point   8.2.  The distribution function of the system of two random variables to the area   8.2.  The distribution function of the system of two random variables denote by symbol   8.2.  The distribution function of the system of two random variables .

The probability of hitting a random point in a given area is expressed most simply in the case when this area is a rectangle with sides parallel to the coordinate axes.

Express through the distribution function of the system the probability of hitting a random point   8.2.  The distribution function of the system of two random variables into a rectangle   8.2.  The distribution function of the system of two random variables limited by abscissas   8.2.  The distribution function of the system of two random variables and   8.2.  The distribution function of the system of two random variables and ordinates   8.2.  The distribution function of the system of two random variables and   8.2.  The distribution function of the system of two random variables (Fig. 8.2.5).

In this case, it should be agreed where we will refer the borders of the rectangle. In the same way as we did for one random variable, we agree to include in the rectangle   8.2.  The distribution function of the system of two random variables its lower and left borders and not include the upper and right. Then event   8.2.  The distribution function of the system of two random variables will be equivalent to the product of two events:   8.2.  The distribution function of the system of two random variables and   8.2.  The distribution function of the system of two random variables . Express the probability of this event through the distribution function of the system. To do this, look at the plane   8.2.  The distribution function of the system of two random variables four infinite quadrants with vertices at points   8.2.  The distribution function of the system of two random variables ;   8.2.  The distribution function of the system of two random variables ;   8.2.  The distribution function of the system of two random variables and   8.2.  The distribution function of the system of two random variables (Fig. 8.2.6).

  8.2.  The distribution function of the system of two random variables

Fig. 8.2.5. Fig. 8.2.6

Obviously, the probability of hitting the rectangle   8.2.  The distribution function of the system of two random variables equal to the probability of falling into the quadrant   8.2.  The distribution function of the system of two random variables minus the probability of hitting the quadrant   8.2.  The distribution function of the system of two random variables minus the probability of hitting the quadrant   8.2.  The distribution function of the system of two random variables plus probability of hitting the quadrant   8.2.  The distribution function of the system of two random variables (since we twice deducted the probability of falling into this quadrant). From here we obtain a formula expressing the probability of hitting a rectangle through the distribution function of the system:

  8.2.  The distribution function of the system of two random variables . (8.2.2)

Later, when the notion of the distribution density of the system is introduced, we derive a formula for the probability that a random point will fall into a region of arbitrary shape.


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis