Find the distribution function of a random variable X, subject to the normal distribution law:
we make a change in the integral and bring it to the form:
.
Integral is not expressed through elementary functions, but it can be calculated through a special function that expresses a definite integral of the expression or . We express the function through the Laplace function Ф(х):
.
The probability of hitting a random variable X on the site (α, β) is expressed by the formula:
.
Using the last formula, one can estimate the probability of deviation of a normal random variable from its mathematical expectation by a predetermined, arbitrarily small positive value ε:
.
Let , then and . At t=3 we get , i.e. the event that the deviation of a normally distributed random variable from the mathematical expectation will be less than , is practically certain.
This is what three sigma rule: if a random variable is normally distributed, then the absolute value of the deviation of its values from the mathematical expectation does not exceed three times the standard deviation.
A task. Let the diameter of the part manufactured by the workshop be a random variable distributed normally, m = 4.5 cm, cm. Find the probability that the size of the diameter of a part taken at random differs from its mathematical expectation by no more than 1 mm.
Solution. This problem is characterized by the following values of the parameters that determine the desired probability: , 
, F(0.2)=0.0793,
test questions
1. What probability distribution is called uniform?
2. What is the form of the distribution function of a random variable uniformly distributed on the interval [ a; b]?
3. How to calculate the probability of hitting the values of a uniformly distributed random variable in a given interval?
4. How is the exponential distribution of a random variable determined?
5. What is the distribution function of a random variable distributed according to an exponential law?
6. What probability distribution is called normal?
7. What properties does the density of the normal distribution have? How do the parameters of a normal distribution affect the appearance of a normal distribution density graph?
8. How to calculate the probability that the values of a normally distributed random variable will fall within a given interval?
9. How to calculate the probability of deviation of the values of a normally distributed random variable from its mathematical expectation?
10. Formulate the rule of "three sigma"?
11. What are the mathematical expectation, variance and standard deviation of a random variable distributed according to a uniform law on the interval [ a; b]?
12. What are the mathematical expectation, variance and standard deviation of a random variable distributed according to an exponential law with parameter λ?
13. What are the mathematical expectation, variance and standard deviation of a random variable distributed according to the normal law with parameters m and ?
Control tasks
1. Random variable X distributed uniformly on the interval [−3, 5]. Find the distribution density and distribution function X. Plot graphs for both functions. Find the probabilities and . Calculate the mathematical expectation, variance and standard deviation X.
2. Buses of route No. 21 run regularly with an interval of 10 minutes. The passenger leaves at a stop at a random time. We consider a random variable X− waiting time for a bus passenger (in minutes). Find the distribution density and distribution function X. Plot graphs for both functions. Find the probability that the passenger will have to wait for the bus no more than five minutes. Find the average bus waiting time and the variance of the bus waiting time.
3. It is established that the repair time of the VCR (in days) is a random variable X, distributed according to the exponential law. The average repair time for a VCR is 10 days. Find the distribution density and distribution function X. Plot graphs for both functions. Find the probability that it will take at least 11 days to repair the VCR.
4. Plot the density and distribution functions of a random variable X, distributed according to the normal law with parameters m= = − 2 and = 0.2.
FORMS OF SETTING THE LAW OF DISTRIBUTION FOR CONTINUOUS RANDOM VARIABLES
FORMS OF SETTING THE LAW OF DISTRIBUTION OF DISCRETE RANDOM VARIABLES
1). Table (row) of distribution - simplest form setting the law of distribution of discrete random variables.
Since the table lists all possible values of a random variable.
2). Distribution polygon . In the graphical representation of a distribution series in a rectangular coordinate system, all possible values of a random variable are plotted along the abscissa axis, and the corresponding probabilities are plotted along the ordinate axis. Then dots are applied and connected by straight line segments. The resulting figure - the distribution polygon - is also a form of specifying the law of distribution of a discrete random variable.
3). distribution function - the probability that a random variable X will take on a value less than some given x, i.e.
| . |
From a geometric point of view, it can be considered as the probability of hitting a random point X to the section of the numerical axis located to the left of the fixed point X.
2) ; ;
Task 2.1. Random value X- the number of hits on the target with 3 shots (see task 1.5). Build a distribution series, a distribution polygon, calculate the values of the distribution function and build its graph.
Solution:
1) Series of distribution of a random variable X presented in the table
| At | , |
| At | , |
| At | , |
| At | |
| at | . |
Plotting along the abscissa of the value X, and along the y-axis - values and choosing a certain scale, we get a graph of the distribution function (Fig. 2.2). The distribution function of a discrete random variable has jumps (discontinuities) at those points where the random variable X takes on specific values specified in the distribution table. The sum of all jumps in the distribution function is equal to one.
Rice. 2.2 - Discrete value distribution function
1). distribution function .
For a continuous random variable, the distribution function graph (Fig. 2.3) has the form of a smooth curve.
Distribution function properties:
c) if .
Rice. 2.3 - Distribution function of a continuous value
2). Distribution density defined as derivative of the distribution function, i.e.
| . |
Curve depicting the distribution density of a random variable, is called distribution curve (Fig. 2.4).
Density properties:
and those. the density is a non-negative function;
b), i.e. area limited distribution curve and the x-axis is always 1.
If all possible values of the random variable X enclosed within a before b, then the second density property takes the form:
Rice. 2.4 - Distribution curve
In practice, it is often necessary to know the probability that a random variable X will take on a value within some range, such as from a to b. The desired probability for discrete random variable X is determined by the formula
since the probability of any single value of a continuous random variable is equal to zero: .
Probability of hitting a continuous random variable X on the interval (a,b) is also determined by the expression:
Task 2.3. Random value X given by the distribution function
Find the density , as well as the probability that, as a result of the test, the random variable X will take the value enclosed in the interval .
Solution:
2. Probability of hitting a random variable X in the interval is determined by the formula. Taking and , we find
In many problems related to normally distributed random variables, it is necessary to determine the probability that a random variable , obeying the normal law with parameters , falls into the interval from to . To calculate this probability, we use the general formula
where is the distribution function of the quantity .
Let us find the distribution function of a random variable distributed according to the normal law with parameters . The distribution density of the value is:
.
(6.3.2)
From here we find the distribution function
. (6.3.3)
Let us make the change of variable in the integral (6.3.3)
and bring it to the form:
(6.3.4)
The integral (6.3.4) is not expressed in terms of elementary functions, but it can be calculated in terms of a special function that expresses a definite integral of the expression or (the so-called probability integral), for which tables are compiled. There are many varieties of such functions, for example:
;
etc. Which of these functions to use is a matter of taste. We will choose as such a function
. (6.3.5)
It is easy to see that this function is nothing but the distribution function for a normally distributed random variable with parameters .
We agree to call the function a normal distribution function. The appendix (Table 1) shows tables of function values.
Let us express the distribution function (6.3.3) of the quantity with parameters and in terms of the normal distribution function . Obviously,
.
(6.3.6)
Now let's find the probability of hitting a random variable on the segment from to . According to formula (6.3.1)
Thus, we have expressed the probability that a random variable , distributed according to the normal law with any parameters, will fall on the plot in terms of the standard distribution function , corresponding to the simplest normal law with parameters 0.1. Note that the function arguments in formula (6.3.7) have a very simple meaning: there is a distance from the right end of the section to the center of dispersion, expressed in standard deviations; - 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 dispersion center, and negative if to the left.
Like any distribution function, the function has the following properties:
3. - non-decreasing function.
In addition, from the symmetry of the normal distribution with parameters about the origin, it follows that
Using this property, in fact, it would be possible to limit the function tables to only positive values of the argument, but in order to avoid an unnecessary operation (subtraction from one), Table 1 of the appendix provides values for both positive and negative arguments.
In practice, one often encounters the problem of calculating the probability that a normally distributed random variable will fall into an area that is symmetrical about the center of dispersion. Consider such a section of length (Fig. 6.3.1). Let us calculate the probability of hitting this site using the formula (6.3.7):
Taking into account the property (6.3.8) of the function and giving the left side of the formula (6.3.9) a more compact form, we obtain a formula for the probability of a random variable distributed according to the normal law falling into a section symmetric with respect to the scattering center:
.
(6.3.10)
Let's solve the following problem. Let us set aside successive segments of length from the scattering center (Fig. 6.3.2) and calculate the probability that a random variable will fall into each of them. Since the curve of the normal law is symmetrical, it is enough to postpone such segments only in one direction.
According to the formula (6.3.7) we find:
(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 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 dispersions (up to fractions of a percent) fit into the section .
This allows, knowing the standard deviation and the mathematical expectation of a random variable, to approximately indicate the range 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 “rule of three sigma”. The rule of three sigma also implies an approximate method for determining the standard deviation of a random variable: they take the maximum practically possible deviation from the average and divide it by three. Of course, this rough method can only be recommended if there are no other, more accurate ways to determine .
Example 1. A random variable , distributed according to the normal law, is an error in measuring a certain distance. When measuring, a systematic error is allowed in the direction of overestimation by 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 1.6 (m) in absolute value.
Solution. The measurement error is a random variable obeying the normal law with parameters and . We need to find the probability that this quantity falls on the interval from to . By formula (6.3.7) we have:
Using the function tables (Appendix, Table 1), we find:
;
,
Example 2. Find the same probability as in the previous example, but on the condition that there is no systematic error.
Solution. By formula (6.3.10), assuming , we find:
.
Example 3. At a target that looks like a strip (freeway), the width of which is 20 m, shooting is carried out in a direction perpendicular to the freeway. Aiming is carried out along the center line of the highway. The standard deviation in the firing direction is equal to m. There is a systematic error in the firing direction: the undershoot is 3 m. Find the probability of hitting the freeway with one shot.
Dispersion of a normal random variable.
Dispersion random variable is the mathematical expectation of the square of the corresponding centered random variable.
It characterizes the degree of spread of the values of a random variable relative to its mathematical expectation, i.e. value range width.
Calculation formulas:
The dispersion can be calculated in terms of the second initial moment:
(6.10)
The dispersion of a random variable characterizes the degree of dispersion (scatter) of the values of a random variable relative to its mathematical expectation. The dispersion of SW (both discrete and continuous) is a non-random (constant) quantity.
The variance of SW has the dimension of the square of a random variable. For clarity, the scattering characteristics use a quantity whose dimension coincides with that of the SW.
Standard deviation (RMS) SW X is called characteristic
. (6.11)
RMS is measured in the same physical units as SW and characterizes the width of the range of SW values.
Dispersion Properties
Dispersion constant With equals zero.
Proof: by definition of variance
When added to a random variable X non-random value With its variance does not change.
D[X+c] = D[X].
Proof: by definition of variance
(6.12)
3. When multiplying a random variable X by a random amount With its variance is multiplied by since 2.
Proof: by definition of variance
. (6.13)
For the standard deviation, this property has the form:
(6.14)
Indeed, when ½C½>1, the value of cX has possible values (in absolute value) greater than the value of X. Therefore, these values are scattered around the mathematical expectation M[cX] is greater than possible values X around M[X], i.e. 
. If 0<½с½<1, то 
.
Rule 3s. For most values of a random variable, the absolute value of its deviation from the mathematical expectation does not exceed three times the standard deviation, or, in other words, almost all CV values are in the interval:
[ m - 3s; m + 3 s; ].(6.15)
Probability of falling into a given interval of a normal random variable
As already established, the probability that a continuous random variable will take a value belonging to the interval is equal to a certain integral of the distribution density, taken within the appropriate limits: 
.
For a normally distributed random variable, respectively, we obtain: 
.
Let's transform the last expression by introducing a new variable 
. Therefore, the exponent of the expression under the integral is converted to: 
.
To replace a variable in a definite integral, it is still necessary to replace the differential and the limits of integration, having previously expressed the variable from the replacement formula: 
;
;
is the lower limit of integration;
is the upper limit of integration;
(to find the limits of integration with respect to the new variable, and are the limits of integration with respect to the old variable were substituted into the variable change formula).
Substitute everything in the last of the formulas for finding the probability: 
where 
is the Laplace function.
Conclusion: the probability that a normally distributed random variable will take a value belonging to the interval is equal to: 
,
where is the mathematical expectation, is the standard deviation of the given random variable.
23. Chi-squared distributions, Student and Fisher
The normal distribution defines three distributions that are now commonly used in statistical data processing. In the following sections of the book, these distributions are encountered many times.
Pearson distribution (chi - square) - distribution of a random variable
where random variables X 1 , X 2 ,…, X n are independent and have the same distribution N(0.1). In this case, the number of terms, i.e. n, is called the "number of degrees of freedom" of the chi-squared distribution.
The chi-square distribution is used in estimating variance (using a confidence interval), in testing hypotheses of agreement, homogeneity, independence, primarily for qualitative (categorized) variables that take on a finite number of values, and in many other tasks of statistical data analysis.
Distribution t Student is the distribution of a random variable
where random variables U and X independent, U has a standard normal distribution N(0,1) and X– distribution chi – square with n degrees of freedom. Wherein n is called the "number of degrees of freedom" of the Student's distribution.
Student's distribution was introduced in 1908 by the English statistician W. Gosset, who worked at a beer factory. Probabilistic-statistical methods were used to make economic and technical decisions at this factory, so its management forbade V. Gosset to publish scientific articles under his own name. In this way, a trade secret was protected, "know-how" in the form of probabilistic-statistical methods developed by W. Gosset. However, he was able to publish under the pseudonym "Student". The history of Gosset-Student shows that a hundred years ago, the great economic efficiency of probabilistic-statistical methods was obvious to British managers.
Currently, the Student's distribution is one of the most well-known distributions among those used in the analysis of real data. It is used in estimating the mathematical expectation, predictive value and other characteristics using confidence intervals, testing hypotheses about the values of mathematical expectations, regression coefficients, hypotheses of sample homogeneity, etc. .
Where 
- integral Laplace function, is given in a table.
From the properties of a definite integral Ф (- X)= - F( X), i.e. function Ф( X) is odd.
From this the following (derivative) formulas are derived:
Assuming: a) d=s
Three Sigma Rule (3s): it is almost certain that in a single test, the deviation of a normally distributed random variable from its mathematical expectation does not exceed three times the standard deviation.
A task: It is assumed that the mass of mirror carps caught in the pond is a random variable X, which has a normal distribution with mathematical expectation a\u003d 375 g. and standard deviation s \u003d 25 g. It is required to determine:
A) The probability that the mass of a randomly caught carp will be at least a=300 g and not more than b=425 g.
B) The probability that the deviation of the specified mass from the average value (mathematical expectation) in absolute value will be less than d = 40 g.
C) Using the rule of three sigma, find the minimum and maximum limits of the estimated mass of mirror carps.
Solution:
BUT) 
Conclusion: Approximately 98% of carp swimming in the pond weigh at least 300 g and not more than 425 g.
B) 
Conclusion: Approximately 89% have a mass of a-d= 375- 40 = 335 to a+ d \u003d 375 + 40 \u003d 415 g.
C) According to the rule of three sigma:
Conclusion: The mass of almost all carps (approximately 100%) lies in the range from 300 to 450 grams.
Tasks for independent solution
1. The shooter hits the target with a probability of 0.8. What is the probability that with three shots the target will be hit exactly twice? At least twice?
2. There are four children in the family. Taking the birth of a boy and a girl as equally likely events, estimate the probability that there are two girls in the family. Three girls and one boy. Compose a distribution law for a random variable X corresponding to the possible number of girls in the family. Calculate characteristics: M(X), s.
3. A die is tossed three times. What is the probability that a "6" will come up once? Not more than once?
4. Random value X uniformly distributed over the interval. What is the probability that a random variable X falls within the interval ?
5. It is assumed that the growth of people (for definiteness - adults, men) living in a certain area obeys the normal distribution law with mathematical expectation a\u003d 170 cm and standard deviation s \u003d 5 cm. What is the probability that the height of a randomly selected person:
A) will be no more than 180 cm and not less than 165 cm?
B) deviates from the average in absolute value by no more than 10 cm?
C) according to the “three sigma” rule, estimate the minimum and maximum possible height of a person.
test questions
1. How is the Bernoulli formula written? When is it applied?
2. What is the binomial distribution law?
3. What random variable is called uniformly distributed?
4. What form do the integral and differential distribution functions have for a random variable uniformly distributed on the interval [ a, b]?
5. What random variable has a normal distribution law?
6. What does the bell curve look like?
7. How to find the probability of a normally distributed random variable falling into a given interval?
8. How is the Three Sigma Rule formulated?
Introduction to the theory of random processes
random function call a function whose value for each value of the independent variable is a random variable.
Random (or stochastic) process is called a random function for which the independent variable is time t.
In other words, a random process is a random variable that changes over time. random process X(t) on is a definite curve, it is a set or a family of definite curves x i (t) (i= 1, 2, …, n) obtained as a result of individual experiments. Each curve in this set is called implementation (or trajectory) random process.
The cross section of a random process called a random variable X(t 0) corresponding to the value of the random process at some fixed time t = t0.