Crossword is a value characterized only by a numerical value. Quantities that are completely determined by their numerical value
RANDOM VALUES AND THE LAWS OF THEIR DISTRIBUTION.
Random called a quantity that takes values depending on the combination of random circumstances. Distinguish discrete and random continuous quantities.
Discrete A quantity is called if it takes a countable set of values. ( Example: the number of patients at the doctor's office, the number of letters per page, the number of molecules in a given volume).
continuous called a quantity that can take values within a certain interval. ( Example: air temperature, body weight, human height, etc.)
distribution law A random variable is a set of possible values of this quantity and, corresponding to these values, probabilities (or frequencies of occurrence).
EXAMPLE:
Numerical characteristics of random variables.
In many cases, along with the distribution of a random variable or instead of it, information about these quantities can be provided by numerical parameters called numerical characteristics of a random variable . The most commonly used of them:
1 .Expected value - (average value) of a random variable is the sum of the products of all its possible values and the probabilities of these values:
2 .Dispersion random variable:
3 .Standard deviation :
THREE SIGMS - if a random variable is distributed according to the normal law, then the deviation of this value from the mean value in absolute value does not exceed three times the standard deviation
Gauss law - normal distribution law
Often there are values distributed over normal law (Gauss' law). main feature : it is the limiting law to which other laws of distribution approach.
A random variable is normally distributed if its probability density looks like:
M(X) - mathematical expectation of a random variable;
- standard deviation.
Probability Density (distribution function) shows how the probability related to the interval changes dx random variable, depending on the value of the variable itself:
Basic concepts of mathematical statistics
Math statistics - a branch of applied mathematics, directly adjacent to the theory of probability. The main difference between mathematical statistics and probability theory is that mathematical statistics does not consider actions on distribution laws and numerical characteristics of random variables, but approximate methods for finding these laws and numerical characteristics based on experimental results.
Basic concepts mathematical statistics are:
General population;
sample;
variation series;
fashion;
median;
percentile,
frequency polygon,
bar chart.
Population - a large statistical population from which some of the objects for research are selected
(Example: the entire population of the region, university students of the city, etc.)
Sample (sample population) - a set of objects selected from the general population.
Variation series - statistical distribution, consisting of variants (values of a random variable) and their corresponding frequencies.
Example:
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X , kg | ||||||||||||
|
m |
x - the value of a random variable (mass of girls aged 10 years);
m - frequency of occurrence.
Fashion – the value of the random variable, which corresponds to the highest frequency of occurrence. (In the example above, 24 kg is the most common value for fashion: m = 20).
Median - the value of a random variable that divides the distribution in half: half of the values are located to the right of the median, half (no more) - to the left.
Example:
1, 1, 1, 1, 1. 1, 2, 2, 2, 3 , 3, 4, 4, 5, 5, 5, 5, 6, 6, 7 , 7, 7, 7, 7, 7, 8, 8, 8, 8, 8 , 8, 9, 9, 9, 10, 10, 10, 10, 10, 10
In the example, we observe 40 values of a random variable. All values are arranged in ascending order, taking into account the frequency of their occurrence. It can be seen that 20 (half) of the 40 values are located to the right of the selected value 7. So 7 is the median.
To characterize the scatter, we find the values that were not higher than 25 and 75% of the measurement results. These values are called the 25th and 75th percentiles . If the median bisects the distribution, then the 25th and 75th percentiles are cut off from it by a quarter. (The median itself, by the way, can be considered the 50th percentile.) As you can see from the example, the 25th and 75th percentiles are 3 and 8, respectively.
use discrete (point) statistical distribution and continuous (interval) statistical distribution.
For clarity, statistical distributions are depicted graphically in the form frequency polygon or - histograms .
Frequency polygon - a broken line, the segments of which connect points with coordinates ( x 1 , m 1 ), (x 2 , m 2 ), ..., or for polygon of relative frequencies - with coordinates ( x 1 ,R * 1 ), (x 2 ,R * 2 ), ...(Fig.1).
mm i / nf(x)
x x
Fig.1 Fig.2
Frequency histogram - a set of adjacent rectangles built on one straight line (Fig. 2), the bases of the rectangles are the same and equal dx , and the heights are equal to the ratio of frequency to dx , or R * to dx (probability density).
Example:
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x, kg | ||||||||||||||||||
71, Numerical characteristics of random variables are widely used in practice for calculating reliability indicators. In many questions of practice, there is no need to completely, exhaustively characterize a random variable. It is often sufficient to indicate only numerical parameters that to some extent characterize the essential features of the distribution of a random variable, for example: mean , near which the possible values of the random variable are grouped; number characterizing the dispersion of a random variable relative to the average value, etc. Numerical parameters that allow expressing in a compressed form the most significant features of a random variable are called numerical characteristics of a random variable.
a) b)
Rice. 11 Definition of expectation
Numerical characteristics of random variables used in reliability theory are given in Table. one.
72,Expectation(mean value) of a continuous random variable whose possible values belong to the interval 
, is a definite integral (Fig., 11, b)
. (26)
The mathematical expectation can be expressed in terms of the complement of the integral function. To do this, we substitute (11) into (26) and integrate by parts the resulting expression
, (27)
because 
and 
, then
. (28)
For non-negative random variables whose possible values belong to the interval , formula (28) takes the form
. (29)
i.e., the mathematical expectation of a non-negative random variable whose possible values belong to the interval , is numerically equal to the area under the graph of the complement of the integral function (Fig., 11, a).
73,Mean time to first failure according to statistical information is determined by the formula
, (30)
where is the time to first failure i-th object; N- number of tested objects.
Similarly, the average resource, average service life, average recovery time, average shelf life are determined.
74, Scattering of a random variable around its expected value evaluated using standard deviation dispersion(RMS) and coefficient of variation.
The dispersion of a continuous random variable X is the mathematical expectation of the squared deviation of the random variable from its mathematical expectation and is calculated by the formula
. (31)
Dispersion has the dimension of the square of a random variable, which is not always convenient.
75,Standard deviation random variable is the square root of the variance and has the dimension of a random variable
. (32)
76,Coefficient of variation is a relative indicator of the dispersion of a random variable and is defined as the ratio of the standard deviation to the mathematical expectation
. (33)
77, Gamma - percentage value of a random variable- the value of the random variable corresponding to the given probability 
that the random variable takes on a value greater than
. (34)
78, Gamma - the percentage value of a random variable can be determined by the integral function, its complement and differential function (Fig. 12). The gamma percentage value of a random variable is the probability quantile (Fig. 12, a)
. (35)
Reliability theory uses gamma percentage value of resource, service life and shelf life(Table 1). Gamma percentage is called resource, service life, shelf life, which has (and exceeds) a percentage of objects of a given type.
a) b)
Fig.12 Determining the gamma percentage value of a random variable
Gamma percent resource characterizes durability at the selected level 
probability of non-destruction. The gamma-percentage resource is assigned taking into account the responsibility of the objects. For example, for rolling bearings, a 90% resource is most often used, for bearings of the most critical objects, a 95% resource and above are chosen, bringing it closer to 100% if the failure is life-threatening.
79, Median of a random variable is its gamma percentage value at 
. For the median 
it is equally likely that the random variable will be T more or less than it, i.e. .
Geometrically, the median is the abscissa of the intersection point of the integral distribution function and its complement (Fig. 12, b). The median can be interpreted as the abscissa of the point at which the ordinate of the differential function bisects the area bounded by the distribution curve (Fig., 12, in).
The median of a random variable is used in the theory of reliability as a numerical characteristic of the resource, service life, shelf life (Table 1).
There is a functional relationship between the reliability indicators of objects. Knowledge of one of the functions 
allows you to determine other indicators of reliability. A summary of the relationships between reliability indicators is given in Table. 2.
Table 2. Functional relationship between reliability indicators
When solving many practical problems, it is not always necessary to characterize a random variable completely, i.e., to determine the laws of distribution. In addition, the construction of a function or a series of distributions for a discrete, and density - for a continuous random variable is cumbersome and unnecessary.
Sometimes it is enough to indicate individual numerical parameters that partially characterize the features of the distribution. It is necessary to know some average value of each random variable, around which its possible value is grouped, or the degree of dispersion of these values relative to the average, etc.
The characteristics of the most significant features of the distribution are called numerical characteristics random variable. With their help, the solution of many probabilistic problems is facilitated without determining the laws of distribution for them.
The most important characteristic of the position of a random variable on the real axis is expected value M[X]= a, which is sometimes called the mean value of the random variable. For discrete random variable X with possible values x 1 , x 2 , … , x n and probabilities p 1 , p 2 ,… , p n it is determined by the formula
Given that =1, we can write
In this way, mathematical expectation A discrete random variable is the sum of the products of its possible values and their probabilities. The arithmetic mean of the observed values of a random variable with a large number of experiments approaches its mathematical expectation.
For continuous random variable X mathematical expectation is not determined by the sum, but integral
where f(x) - distribution density of quantity x.
Mathematical expectation does not exist for all random variables. For some of them, the sum, or integral, diverges, and therefore there is no expectation. In these cases, for reasons of accuracy, one should limit the range of possible changes in the random variable x, for which the sum, or integral, will converge.
In practice, such characteristics of the position of a random variable as mode and median are also used.
Random fashionits most probable value is called. In the general case, the mode and the mathematical expectation do not coincide.
Median of a random variableX is its value, with respect to which it is equally likely to obtain a larger or smaller value of a random variable, i.e., this is the abscissa of the point at which the area bounded by the distribution curve is divided in half. For a symmetrical distribution, all three characteristics are the same.
In addition to the mathematical expectation, mode and median, other characteristics are also used in probability theory, each of which describes a certain property of the distribution. For example, numerical characteristics that characterize the dispersion of a random variable, i.e., showing how closely its possible values are grouped around the mathematical expectation, are the variance and the standard deviation. They significantly complement the random variable, since in practice there are often random variables with equal mathematical expectations, but different distributions. When determining the scattering characteristics, the difference between the random variable X and its mathematical expectation, i.e.
where a = M[X] - expected value.
This difference is called centered random variable, corresponding value x, and denoted :
Variance of a random variable is the mathematical expectation of the square of the deviation of a value from its mathematical expectation, i.e.:
D[ X]=M[( X-a) 2 ], or
D[ X]=M[ 2 ].
The variance of a random variable is a convenient characteristic of dispersion and dispersion of the values of a random variable around its mathematical expectation. However, it is devoid of visibility, since it has the dimension of the square of a random variable.
For a visual characterization of scattering, it is more convenient to use a quantity whose dimension coincides with that of a random variable. This value is standard deviation random variable that is the positive square root of its variance.
Mathematical expectation, mode, median, variance, standard deviation - the most commonly used numerical characteristics of random variables. When solving practical problems, when it is impossible to determine the distribution law, an approximate description of a random variable is its numerical characteristics, expressing some property of the distribution.
In addition to the main characteristics of the distribution of the center (expectation) and dispersion (dispersion), it is often necessary to describe other important characteristics of the distribution - symmetry and sharpness, which can be represented using the distribution moments.
The distribution of a random variable is completely given if all its moments are known. However, many distributions can be fully described using the first four moments, which are not only parameters describing distributions, but are also important in the selection of empirical distributions, that is, by calculating the numerical values of the moments for a given statistical series and using special graphs, one can determine the distribution law.
In probability theory, two types of moments are distinguished: initial and central.
The initial moment of the kth order random variable T is called the mathematical expectation of the quantity X k , i.e.
Therefore, for a discrete random variable, it is expressed by the sum
and for continuous - integral
Among the initial moments of a random variable, the moment of the first order, which is the mathematical expectation, is of particular importance. Higher-order initial moments are mainly used to calculate the central moments.
The central moment of the kth order random variable is called the mathematical expectation of the variable ( X - M [X])k
where a = M[X].
For a discrete random variable, it is expressed by the sum
a for continuous - integral
Among the central moments of a random variable, the second order central moment, which represents the variance of the random variable.
The first order central moment is always zero.
Third initial moment characterizes the asymmetry (skewness) of the distribution and, according to the results of observations for discrete and continuous random variables, is determined by the corresponding expressions:
Since it has the dimension of a cube of a random variable, in order to obtain a dimensionless characteristic, m 3 divided by the standard deviation to the third power
The resulting value is called the asymmetry coefficient and, depending on the sign, characterizes the positive ( As> 0) or negative ( As< 0) the skewness of the distribution (Fig. 2.3).
"Units of measurement of physical quantities" - The absolute error is equal to half the scale division of the measuring instrument. Micrometer. The result is obtained directly with the measuring device. Box length: 4 cm short, 5 cm over. For each physical quantity there are corresponding units of measurement. Watch. Relative error.
“Length values” - 2. What quantities can be compared with each other: 2. Explain why the following problem is solved using addition: 2. Justify the choice of action when solving the problem. How many packages did you get? How many pens are in three of these boxes? Dresses were sewn from 12 m of fabric, spending 4 m each. How many dresses were sewn?
"Physical quantities" - The boundaries separating physics and other natural sciences are historically conditional. The result of any measurement always contains some error. New topic. Speed. Phone interaction. Physical laws are presented in the form of quantitative ratios expressed in the language of mathematics. Measurement error.
“Number as a result of measuring a value” - “Number as a result of measuring a value” math lesson in grade 1. Measuring the length of a segment with a yardstick.
"Numbers and quantities" - Acquaintance with the concept of mass. Comparison of masses without measurements. Roman written numbering. Capacity. The student will learn: Numbers and quantities (30 hours) Coordinate ray The concept of a coordinate ray. Planned subject results in the section "Numbers and quantities" in grade 2. The general principle of the formation of cardinal numbers within the studied numbers.
"Magnitude of demand" - Causes of changes in demand. The DD curve obtained on the chart (from the English demand - "demand") is called the demand curve. Elastic demand (Epd>1). The amount of demand. Factors affecting demand. The dependence of the quantity demanded on the price level is called the scale of demand. Absolutely inelastic demand (Epd=0).
Expected value. mathematical expectation discrete random variable X, which takes a finite number of values Xi with probabilities Ri, is called the sum:
mathematical expectation continuous random variable X is called the integral of the product of its values X on the probability distribution density f(x):
(6b)
Improper integral (6 b) is assumed to be absolutely convergent (otherwise we say that the expectation M(X) does not exist). The mathematical expectation characterizes mean random variable X. Its dimension coincides with the dimension of a random variable.
Properties of mathematical expectation:
Dispersion. dispersion random variable X number is called:
The dispersion is scattering characteristic values of a random variable X relative to its average value M(X). The dimension of the variance is equal to the dimension of the random variable squared. Based on the definitions of variance (8) and mathematical expectation (5) for a discrete random variable and (6) for a continuous random variable, we obtain similar expressions for the variance:
(9)
Here m = M(X).
Dispersion properties:
Standard deviation:
(11)
Since the dimension of the standard deviation is the same as that of a random variable, it is more often than the variance used as a measure of dispersion.
distribution moments. The concepts of mathematical expectation and variance are special cases of a more general concept for the numerical characteristics of random variables - distribution moments. The distribution moments of a random variable are introduced as mathematical expectations of some simple functions of a random variable. So, the moment of order k relative to the point X 0 is called expectation M(X–X 0 )k. Moments relative to the origin X= 0 are called initial moments and are marked:
(12)
The initial moment of the first order is the distribution center of the considered random variable:
(13)
Moments relative to distribution center X= m called central points and are marked:
(14)
From (7) it follows that the central moment of the first order is always equal to zero:
The central moments do not depend on the origin of the values of the random variable, since with a shift by a constant value FROM its center of distribution is shifted by the same value FROM, and the deviation from the center does not change: X – m = (X – FROM) – (m – FROM).
Now it is obvious that dispersion- this is second order central moment:
Asymmetry. Central moment of the third order:
(17)
serves to evaluate distribution skewness. If the distribution is symmetrical about the point X= m, then the central moment of the third order will be equal to zero (as well as all central moments of odd orders). Therefore, if the central moment of the third order is different from zero, then the distribution cannot be symmetric. The magnitude of the asymmetry is estimated using a dimensionless asymmetry coefficient:
(18)
The sign of the asymmetry coefficient (18) indicates right-sided or left-sided asymmetry (Fig. 2).
Rice. 2. Types of asymmetry of distributions.
Excess. Central moment of the fourth order:
(19)
serves to evaluate the so-called kurtosis, which determines the degree of steepness (pointiness) of the distribution curve near the distribution center with respect to the normal distribution curve. Since for a normal distribution, the quantity taken as kurtosis is:
(20)
On fig. 3 shows examples of distribution curves with different values of kurtosis. For a normal distribution E= 0. Curves that are more peaked than normal have positive kurtosis, and curves with more flat peaks have negative kurtosis.
Rice. 3. Distribution curves with different degrees of steepness (kurtosis).
Higher-order moments in engineering applications of mathematical statistics are usually not used.
Fashion
discrete random variable is its most probable value. Fashion continuous a random variable is its value at which the probability density is maximum (Fig. 2). If the distribution curve has one maximum, then the distribution is called unimodal. If the distribution curve has more than one maximum, then the distribution is called polymodal. Sometimes there are distributions whose curves have not a maximum, but a minimum. Such distributions are called antimodal. In the general case, the mode and the mathematical expectation of a random variable do not coincide. In a particular case, for modal, i.e. having a mode, a symmetric distribution, and provided that there is a mathematical expectation, the latter coincides with the mode and the center of symmetry of the distribution.
Median random variable X is its meaning Me, for which equality holds: i.e. it is equally likely that the random variable X will be less or more Me. Geometrically median is the abscissa of the point at which the area under the distribution curve is divided in half (Fig. 2). In the case of a symmetric modal distribution, the median, mode, and mean are the same.
