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6.3. The probability of hitting a random variable in a given area. Normal distribution function

Lecture



The probability of hitting a random variable, subordinate to the normal law, on a given area. Normal distribution function

In many problems associated with normally distributed random variables, it is necessary to determine the probability of a random variable falling.   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function subject to normal law with parameters   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function on the plot from   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function before   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function . To calculate this probability, we use the general formula

  6.3.  The probability of hitting a random variable in a given area.  Normal distribution function , (6.3.1)

Where   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function - size distribution function   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function .

Find the distribution function   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function random variable   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function distributed according to normal law with parameters   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function . Distribution density   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function equals:

  6.3.  The probability of hitting a random variable in a given area.  Normal distribution function . (6.3.2)

From here we find the distribution function

  6.3.  The probability of hitting a random variable in a given area.  Normal distribution function . (6.3.3)

We make in the integral (6.3.3) a change of variable

  6.3.  The probability of hitting a random variable in a given area.  Normal distribution function

and bring it to mind:

  6.3.  The probability of hitting a random variable in a given area.  Normal distribution function (6.3.4)

The integral (6.3.4) is not expressed in terms of elementary functions, but it can be calculated through a special function expressing a definite integral of the expression   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function or   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function (the so-called probability integral) for which the tables are compiled. There are many types of such functions, for example:

  6.3.  The probability of hitting a random variable in a given area.  Normal distribution function ;   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function

etc. Which of these functions to use is a matter of taste. We will choose as such a function

  6.3.  The probability of hitting a random variable in a given area.  Normal distribution function . (6.3.5)

It is easy to see that this function is nothing but a distribution function for a normally distributed random variable with parameters   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function .

Let's call the function   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function normal distribution function. The appendix (Table 1) contains tables of function values.   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function .

Express the distribution function (6.3.3) values   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function with parameters   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function and   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function through the normal distribution function   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function . Obviously

  6.3.  The probability of hitting a random variable in a given area.  Normal distribution function . (6.3.6)

Now we find the probability of hitting a random variable.   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function on the plot from   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function before   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function . According to the formula (6.3.1)

  6.3.  The probability of hitting a random variable in a given area.  Normal distribution function . (6.3.7)

Thus, we have expressed the probability that a random variable will hit the section.   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function distributed according to the normal law with any parameters through the standard distribution function   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function corresponding to the simplest normal law with parameters 0,1. Note that the function arguments   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function in the formula (6.3.7) have a very simple meaning:   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function there is a distance from the right end of the plot   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function to the center of dispersion, expressed in standard deviations;   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function - the same distance for the left end of the section, and this distance is considered positive if the end is located to the right of the center of dispersion, and negative if it is to the left.

Like any distribution function, the function   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function has properties:

one.   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function .

2   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function .

3   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function - non-decreasing function.

In addition, from the symmetry of the normal distribution with parameters   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function relative to the origin it follows that

  6.3.  The probability of hitting a random variable in a given area.  Normal distribution function . (6.3.8)

Using this property, strictly speaking, it would be possible to limit the function tables   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function only positive values ​​of the argument, but to avoid unnecessary operation (subtraction from one), the values ​​in table 1 of the annex are given   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function for both positive and negative arguments.

  6.3.  The probability of hitting a random variable in a given area.  Normal distribution function

Fig. 6.3.1.

In practice, the problem of calculating the probability of a normally distributed random variable in a region symmetrical about the center of dispersion is often encountered.   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function . Consider this length   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function (fig. 6.3.1). We calculate the probability of hitting this area by the formula (6.3.7):

  6.3.  The probability of hitting a random variable in a given area.  Normal distribution function . (6.3.9)

Given the property (6.3.8) of the function   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function and giving the left side of the formula (6.3.9) a more compact form, we obtain the formula for the probability of a random variable falling on the area symmetric about the center of dispersion according to the normal law:

  6.3.  The probability of hitting a random variable in a given area.  Normal distribution function . (6.3.10)

We solve the following problem. Set aside from the center of dispersion   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function consecutive lengths   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function (Fig. 6.3.2) and calculate the probability of hitting a random variable   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function in each of them. Since the curve of the normal law is symmetric, it is enough to postpone such segments only in one direction.

  6.3.  The probability of hitting a random variable in a given area.  Normal distribution function

Fig. 6.3.2.

By the formula (6.3.7) we find:

  6.3.  The probability of hitting a random variable in a given area.  Normal distribution function (6.3.11)

As can be seen from these data, the probabilities of hitting each of the following segments (fifth, sixth, etc.) with an accuracy of up to 0.001 are equal to zero.

Rounding the probabilities of hitting the segments to 0.01 (up to 1%), we get three numbers that are easy to remember:

0.34; 0.14; 0.02.

The sum of these three values ​​is 0.5. This means that for a normally distributed random variable, all the dispersions (up to fractions of a percent) fit into the area   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function .

This allows, knowing the mean square deviation and the mathematical expectation of a random variable, to roughly indicate the interval of its practically possible values. This method of estimating the range of possible values ​​of a random variable is known in mathematical statistics as the “three sigma rule”. The approximate method of determining the standard deviation of a random variable also follows from the rule of three sigma: take the maximum practically possible deviation from the average and divide it into three. Of course, this rude reception can be recommended only if there are no other, more accurate ways to determine   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function .

Example 1. Random variable   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function , distributed according to the normal law, is a measurement error of a certain distance. When measuring a systematic error in the direction of overestimation of 1.2 (m); the standard deviation of the measurement error is 0.8 (m). Find the probability that the deviation of the measured value from the true value does not exceed the absolute value of 1.6 (m).

Decision. Measurement error is random   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function subject to normal law with parameters   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function and   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function . It is necessary to find the probability of hitting this quantity from   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function before   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function . By the formula (6.3.7) we have:

  6.3.  The probability of hitting a random variable in a given area.  Normal distribution function .

Using function tables   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function (Appendix, Table 1), we find:

  6.3.  The probability of hitting a random variable in a given area.  Normal distribution function ;   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function ,

from where

  6.3.  The probability of hitting a random variable in a given area.  Normal distribution function .

Example 2. Find the same probability as in the previous example, but under the condition that there is no systematic error.

Decision. According to the formula (6.3.10), assuming   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function , we will find:

  6.3.  The probability of hitting a random variable in a given area.  Normal distribution function .

Example 3. The target, having the form of a strip (freeway), whose width is equal to 20 m, is fired in the direction perpendicular to the freeway. Aiming is on the middle line of the highway. The standard deviation in the direction of shooting is   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function m. There is a systematic error in the direction of shooting: 3 m undershoot. Find the probability of hitting the freeway with one shot.

  6.3.  The probability of hitting a random variable in a given area.  Normal distribution function

Fig. 6.3.3.

Decision. Select the origin at any point on the middle line of the motorway (Fig. 6.3.3) and direct the x-axis perpendicular to the motorway. Hitting or missing a projectile on a freeway is determined by the value of only one coordinate of the drop point   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function (other coordinate   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function we are indifferent). Random value   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function distributed according to normal law with parameters   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function ,   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function . The projectile hit the freeway corresponds to the hit value   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function on the plot from   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function before   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function . Applying the formula (6.3.7), we have:

  6.3.  The probability of hitting a random variable in a given area.  Normal distribution function .

  6.3.  The probability of hitting a random variable in a given area.  Normal distribution function

Fig. 6.3.4.

Example 4. There is a random variable   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function , normally distributed, with a dispersion center   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function (fig. 6.3.4) and some section   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function abscissa axis. What should be the standard deviation   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function random variable   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function in order to probability of hitting   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function on the plot   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function reached the maximum?

Decision. We have:

  6.3.  The probability of hitting a random variable in a given area.  Normal distribution function .

Let's differentiate this function   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function :

  6.3.  The probability of hitting a random variable in a given area.  Normal distribution function ,

but

  6.3.  The probability of hitting a random variable in a given area.  Normal distribution function .

Applying the rule of differentiation of an integral with respect to a variable entering its limit, we obtain:

  6.3.  The probability of hitting a random variable in a given area.  Normal distribution function .

Similarly

  6.3.  The probability of hitting a random variable in a given area.  Normal distribution function .

To find the extremum, we set:

  6.3.  The probability of hitting a random variable in a given area.  Normal distribution function . (6.3.12)

With   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function this expression vanishes and the probability   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function reaches a minimum. Maximum   6.3.  The probability of hitting a random variable in a given area.  Normal distribution function get from the condition

  6.3.  The probability of hitting a random variable in a given area.  Normal distribution function . (6.3.13)

Equation (6.3.13) can be solved numerically or graphically.


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis