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5.2. Distribution function

Lecture



In the previous section, we introduced the distribution series as an exhaustive characteristic (distribution law) of a discontinuous random variable. However, this characteristic is not universal; it exists only for discontinuous random variables. It is easy to verify that such a characteristic cannot be constructed for a continuous random variable. Indeed, a continuous random variable has an infinite number of possible values ​​that completely fill a certain interval (the so-called “countable set”). It is impossible to create a table in which all possible values ​​of such a random variable would be listed. In addition, as we will see later, each individual value of a continuous random variable usually does not have any non-zero probability. Consequently, for a continuous random variable, there is no series of distribution in the sense in which it exists for a continuous quantity. However, different regions of possible values ​​of a random variable are still not equally probable, and for a continuous quantity there is a “probability distribution”, although not in the same sense as for a discontinuous one.

To quantify this probability distribution, it is convenient to use the non-probability of the event.   5.2.  Distribution function and the probability of an event   5.2.  Distribution function where   5.2.  Distribution function - some current variable. The probability of this event obviously depends on   5.2.  Distribution function , there is some function from   5.2.  Distribution function . This function is called the distribution function of the random variable.   5.2.  Distribution function and is denoted by   5.2.  Distribution function :

  5.2.  Distribution function . (5.2.1)

Distribution function   5.2.  Distribution function sometimes also called integral distribution function or integral distribution law.

The distribution function is the most universal characteristic of a random variable. It exists for all random variables: both discontinuous and continuous. The distribution function fully characterizes a random variable from a probabilistic point of view, i.e. is a form of distribution law.

We formulate some general properties of the distribution function.

1. Distribution function   5.2.  Distribution function there is a non-decreasing function of its argument, i.e. at   5.2.  Distribution function   5.2.  Distribution function .

2. At minus infinity, the distribution function is zero:   5.2.  Distribution function .

3. At plus infinity, the distribution function is equal to one:   5.2.  Distribution function .

Without giving a rigorous proof of these properties, we illustrate them with a visual geometric interpretation. For this we will consider a random variable.   5.2.  Distribution function as a random point   5.2.  Distribution function on the Ox axis (fig. 5.2.1), which as a result of experience may take one or another position. Then the distribution function   5.2.  Distribution function there is a chance that a random point   5.2.  Distribution function as a result of the experience will fall to the left of the point   5.2.  Distribution function .

  5.2.  Distribution function

Fig. 5.2.1.

We will increase   5.2.  Distribution function i.e. move point   5.2.  Distribution function right on the x-axis. Obviously, with this, the probability that a random point   5.2.  Distribution function will fall to the left   5.2.  Distribution function , can not decrease; therefore, the distribution function   5.2.  Distribution function with increasing   5.2.  Distribution function can not decrease.

To make sure that   5.2.  Distribution function , we will move the point indefinitely   5.2.  Distribution function left on the x-axis. At the same time hit a random point   5.2.  Distribution function to the left   5.2.  Distribution function in the limit becomes an impossible event; it is natural to assume that the probability of this event tends to zero, i.e.   5.2.  Distribution function .

Similarly, moving the point indefinitely   5.2.  Distribution function right make sure that   5.2.  Distribution function since the event   5.2.  Distribution function becomes in the limit authentic.

Distribution function graph   5.2.  Distribution function in the general case, it is a graph of a non-decreasing function (Fig. 5.2.2), the values ​​of which start from 0 and reach 1, and at certain points the function may have jumps (discontinuities).

  5.2.  Distribution function

Fig. 5.2.2.

Knowing the distribution number of a discontinuous random variable, one can easily construct the distribution function of this quantity. Really,

  5.2.  Distribution function ,

where inequality   5.2.  Distribution function under the sum sign indicates that the summation applies to all those values   5.2.  Distribution function which are smaller   5.2.  Distribution function .

When is the current variable   5.2.  Distribution function passes through any of the possible values ​​of the discontinuous value   5.2.  Distribution function , the distribution function changes abruptly, and the magnitude of the jump is equal to the probability of this value.

Example 1. One experience is performed in which an event may or may not appear.   5.2.  Distribution function . Event probability   5.2.  Distribution function equal to 0.3. Random value   5.2.  Distribution function - the number of occurrences   5.2.  Distribution function in experience (characteristic random variable of event   5.2.  Distribution function ). Build its distribution function.

Decision. Distribution range   5.2.  Distribution function has the form:

  5.2.  Distribution function

Construct the distribution function of   5.2.  Distribution function :

1) at   5.2.  Distribution function

  5.2.  Distribution function ;

2) at   5.2.  Distribution function

  5.2.  Distribution function ;

3) at   5.2.  Distribution function

  5.2.  Distribution function .

The graph of the distribution function is shown in Fig. 5.2.3. At break points function   5.2.  Distribution function accepts values ​​marked by dots in the drawing (the function is continuous on the left).

  5.2.  Distribution function

Fig. 5.2.3.

Example 2. In the conditions of the previous example, 4 independent experiments are performed. Build the distribution function of the number of occurrences of the event   5.2.  Distribution function .

Decision. Denote   5.2.  Distribution function - the number of occurrences   5.2.  Distribution function in four experiments. This value has a number of distribution

  5.2.  Distribution function

Construct the distribution function of a random variable   5.2.  Distribution function :

1) at   5.2.  Distribution function   5.2.  Distribution function ;

2) at   5.2.  Distribution function   5.2.  Distribution function ;

3) at   5.2.  Distribution function   5.2.  Distribution function ;

4) at   5.2.  Distribution function   5.2.  Distribution function ;

5) at   5.2.  Distribution function   5.2.  Distribution function ;

6) at   5.2.  Distribution function   5.2.  Distribution function .

The graph of the distribution function is shown in Fig. 5.2.4.

  5.2.  Distribution function

Fig. 5.2.4.

The distribution function of any discontinuous random variable is always a discontinuous step function whose jumps occur at points corresponding to possible random values ​​of the magnitude and are equal to the probabilities of these values. The sum of all function jumps   5.2.  Distribution function equals one.

As the number of possible values ​​of a random variable increases and the intervals between them jumps, the jumps become larger, and the jumps themselves become smaller; the stepped curve becomes smoother (Fig. 5.2.5); the random value gradually approaches the continuous value, and its distribution function to the continuous function (Fig. 5.2.6).

  5.2.  Distribution function

Fig. 5.2.5.

  5.2.  Distribution function

Fig. 5.2.6.

In practice, the distribution function of a continuous random variable is usually a function that is continuous at all points, as shown in Fig. 5.2.6. However, it is possible to construct examples of random variables, the possible values ​​of which continuously fill a certain gap, but for which the distribution function is not everywhere continuous, but at certain points it suffers a discontinuity (Fig. 5.2.7).

  5.2.  Distribution function

Fig. 5.2.7.

Such random variables are called mixed. As an example of a mixed quantity, we can take the area of ​​damage caused to a target by a bomb whose radius of destructive action is equal to R (Fig. 5.2.8).

  5.2.  Distribution function

Fig. 5.2.8.

The values ​​of this random variable continuously fill the gap from 0 to   5.2.  Distribution function , carried out at the positions of a bomb type I and II, have a certain finite probability, and the values ​​of the distribution function correspond to these values, whereas in intermediate values ​​(position of type III) the distribution function is continuous. Another example of a mixed random variable is the time T of the failure-free operation of the device, tested during the time t. The distribution function of this random variable is continuous everywhere except at the point t.


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis