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7.2. Simple statistical aggregate. Statistical distribution function

Lecture



Suppose we are studying some random variable   7.2.  Simple statistical aggregate.  Statistical distribution function , the distribution law of which is exactly unknown, and it is required to determine this law from experience or experimentally test the hypothesis that the quantity   7.2.  Simple statistical aggregate.  Statistical distribution function subject to one or another law. To this end, over a random variable   7.2.  Simple statistical aggregate.  Statistical distribution function A number of independent experiments (observations) are carried out. In each of these experiments, the random variable   7.2.  Simple statistical aggregate.  Statistical distribution function takes a certain value. The aggregate of the observed values ​​of the quantity is the primary statistical material to be processed, comprehended and scientifically analyzed. Such a set is called a “simple statistical set” or a “simple statistical series”. Usually a simple statistical aggregate is drawn up in the form of a table with one entry, in the first column of which is the number of experience   7.2.  Simple statistical aggregate.  Statistical distribution function , and in the second - the observed value of a random variable.

Example 1. Random variable   7.2.  Simple statistical aggregate.  Statistical distribution function - the glide angle of the aircraft at the time of dropping the bomb (the glide angle implies the angle drawn up by the velocity vector and the plane of symmetry of the airplane). Produced 20 bombings, each of which recorded a slip angle   7.2.  Simple statistical aggregate.  Statistical distribution function in thousandths of radians. The results of the observations are summarized in a simple statistical series:

  7.2.  Simple statistical aggregate.  Statistical distribution function

  7.2.  Simple statistical aggregate.  Statistical distribution function

  7.2.  Simple statistical aggregate.  Statistical distribution function

  7.2.  Simple statistical aggregate.  Statistical distribution function

  7.2.  Simple statistical aggregate.  Statistical distribution function

  7.2.  Simple statistical aggregate.  Statistical distribution function

one

2

3

four

five

6

7

-20

-60

-ten

thirty

60

70

-ten

eight

9

ten

eleven

12

13

14

-thirty

120

-100

-80

20

40

-60

15

sixteen

17

18

nineteen

20

-ten

20

thirty

-80

60

70

A simple statistical series is the primary form of recording statistical material and can be processed in various ways. One of the methods of such processing is the construction of a statistical distribution function of a random variable.

The statistical distribution function of a random variable   7.2.  Simple statistical aggregate.  Statistical distribution function called event frequency   7.2.  Simple statistical aggregate.  Statistical distribution function in this statistical material:

  7.2.  Simple statistical aggregate.  Statistical distribution function . (7.2.1)

In order to find the value of the statistical distribution function for a given   7.2.  Simple statistical aggregate.  Statistical distribution function , it is enough to count the number of experiments in which the magnitude   7.2.  Simple statistical aggregate.  Statistical distribution function took a value less than   7.2.  Simple statistical aggregate.  Statistical distribution function and divide by total   7.2.  Simple statistical aggregate.  Statistical distribution function produced experiences.

Example 2. Construct a statistical distribution function for a random variable   7.2.  Simple statistical aggregate.  Statistical distribution function considered in the previous example.

Decision. Since the smallest observed value is   7.2.  Simple statistical aggregate.  Statistical distribution function then   7.2.  Simple statistical aggregate.  Statistical distribution function . Value   7.2.  Simple statistical aggregate.  Statistical distribution function observed once, its frequency is   7.2.  Simple statistical aggregate.  Statistical distribution function ; hence at   7.2.  Simple statistical aggregate.  Statistical distribution function   7.2.  Simple statistical aggregate.  Statistical distribution function has a jump equal to   7.2.  Simple statistical aggregate.  Statistical distribution function . In the interval from   7.2.  Simple statistical aggregate.  Statistical distribution function before   7.2.  Simple statistical aggregate.  Statistical distribution function function   7.2.  Simple statistical aggregate.  Statistical distribution function has the meaning   7.2.  Simple statistical aggregate.  Statistical distribution function ; at the point   7.2.  Simple statistical aggregate.  Statistical distribution function function jump occurs   7.2.  Simple statistical aggregate.  Statistical distribution function on   7.2.  Simple statistical aggregate.  Statistical distribution function as the value   7.2.  Simple statistical aggregate.  Statistical distribution function observed twice, etc.

The graph of the statistical distribution function of the magnitude is shown in Fig.7.2.1.

  7.2.  Simple statistical aggregate.  Statistical distribution function

Fig. 7.2.1

The statistical distribution function of any random variable, continuous or continuous, is a discontinuous step function whose jumps correspond to the observed values ​​of the random variable and are equal in magnitude to the frequencies of these values. If every single value of a random variable   7.2.  Simple statistical aggregate.  Statistical distribution function was observed only once, the jump in the statistical distribution function in each observed value is   7.2.  Simple statistical aggregate.  Statistical distribution function where   7.2.  Simple statistical aggregate.  Statistical distribution function - the number of observations.

With an increase in the number of experiences   7.2.  Simple statistical aggregate.  Statistical distribution function , according to the Bernoulli theorem, for any   7.2.  Simple statistical aggregate.  Statistical distribution function event frequency   7.2.  Simple statistical aggregate.  Statistical distribution function approaches (converges in probability) to the probability of this event. Consequently, with increasing   7.2.  Simple statistical aggregate.  Statistical distribution function statistical distribution function   7.2.  Simple statistical aggregate.  Statistical distribution function approaches (converges in probability) to the true distribution function   7.2.  Simple statistical aggregate.  Statistical distribution function random variable   7.2.  Simple statistical aggregate.  Statistical distribution function .

If a   7.2.  Simple statistical aggregate.  Statistical distribution function - continuous random variable, then with increasing number of observations   7.2.  Simple statistical aggregate.  Statistical distribution function the number of function jumps   7.2.  Simple statistical aggregate.  Statistical distribution function increases, most jumps decrease and the function graph   7.2.  Simple statistical aggregate.  Statistical distribution function unboundedly approaching a smooth curve   7.2.  Simple statistical aggregate.  Statistical distribution function - size distribution functions   7.2.  Simple statistical aggregate.  Statistical distribution function .

In principle, the construction of a statistical distribution function already solves the problem of describing experimental material. However, with a large number of experiments   7.2.  Simple statistical aggregate.  Statistical distribution function construction   7.2.  Simple statistical aggregate.  Statistical distribution function the above method is very time consuming. In addition, it is often convenient, in the sense of clarity, to use other characteristics of statistical distributions that are similar to non-distribution functions.   7.2.  Simple statistical aggregate.  Statistical distribution function , and density   7.2.  Simple statistical aggregate.  Statistical distribution function . With such ways of describing statistical data we will get acquainted in the next paragraph.


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis