Theoretical basis For mathematical statistics, probability theory is used, which studies patterns random phenomena in an abstract form. Based on these patterns, models or laws of distribution of random values ​​are developed.

The law of distribution of a discrete quantity is a task of its probabilities possible values X = xi. Continuous distribution law random variable represented as a distribution function of X values< x i , т. е. в integral form and in the form of distribution density. Probability separate meaning continuous random variable is equal to 0, and the probability of values ​​included in a given gradation is equal to the increment of the distribution function in the area occupied by this gradation Δx.

Each theoretical distribution has characteristics similar to those of statistical distributions (expectation M, variance D, coefficients of variation, skewness and kurtosis). These or other constants associated with them are called distribution parameters.

Finding a theoretical distribution that matches the empirical one, or “leveling” it, is one of the important tasks climatological processing. If a theoretical distribution is found and successfully found, then the climatologist receives not only a convenient form of representation of the value being studied, which can be included in machine calculations, but also the ability to calculate characteristics not directly contained in the original series, as well as to identify certain patterns. Thus, the extremes observed at the point are certainly of interest. However, their appearance in the available sample is largely random, so they are poorly mapped and sometimes differ significantly at neighboring stations. If, with the help of the found distributions, we determine the extreme characteristics of a certain security, then they are largely free from the mentioned shortcomings and are therefore more representative. It is on the calculated extrema that various regulatory requirements. Therefore, special attention should be paid to finding a theoretical distribution and checking its correctness.

Distribution parameters can be determined different ways, the most accurate, but at the same time complex, is the maximum likelihood method. In climatological practice, the method of moments is used.

Statistical characteristics are considered as estimates of distribution parameters characterizing the general population of values ​​of a given random variable.

The moment method for determining parameter estimates is as follows. Expected value, theoretical coefficients skewness and kurtosis are simply replaced by the empirical mean and empirical coefficients; The theoretical variance is equal to the empirical variance multiplied by . If the parameters are functions of moments, then they are calculated from empirical moments.

Let's look at some probabilistic models, often used in climatology.

For discrete random variables, binomial and Poisson distributions (simple and complex) are used.

The binomial distribution (Bernoulli) arises as a result of repetition under constant conditions of the same test, which has two outcomes: the occurrence or non-occurrence of an event (in climatology, for example, the absence or presence of an event on every day of the year or month).

Random discrete quantity is understood here as the number of cases of the occurrence of some random event (phenomenon) out of n possible cases and can take values ​​0, 1, 2, ..., n.

Analytical expression The binomial distribution law has the form (5.1)

The law determines the probability that an event with probability p will occur x times in n trials. For example, in climatology, a day can be either with a phenomenon or without a phenomenon (with fog, with a certain amount of precipitation, air temperature of certain gradations, etc.). In all these cases, two outcomes are possible, and the question of how many times an event (for example, a day with fog) will be observed can be answered using the binomial law (5.1). In this case, p is taken equal to p*, i.e., relative frequency - the ratio of the number of cases with a phenomenon to the total number of cases (formula (2.3)).

For example, if the number of days with fog in August is considered and it is established from a long-term series that on average there are 5 days with fog in August, then the relative frequency (probability) of a day with fog in August (31 days) is equal to

The parameters of the binomial distribution are n and p, which are related to the mathematical expectation (mean value), mean square deviation, coefficients of asymmetry and kurtosis of this distribution by the following expressions:

In Fig. 5.1 shows graphs of the binomial distribution for different parameters n and p.

Let us calculate, for example, using the binomial law, the probability that in August the station will experience three days with fog if the probability of fog formation on any day in August (i.e. the ratio of the average number of days with fog in August to the total number of days in the month ) is 0.16.

Since n = 31, and 1 - p = 0.84, using formula (5.1) we obtain

p(3)=0.1334≈0.13

The limit of the binomial distribution, provided that low-probability events in a long series are considered independent tests(observations) is the Poisson distribution.

A random variable distributed according to Poisson's law can take a number of values ​​forming an infinite sequence of integers 0, 1, 2, ∞ with probability

where λ. -parameter, which is the mathematical expectation of the distribution.

The law determines the probability that a random variable will be observed x times if its average value (mathematical expectation) is equal to λ.

Let us pay attention to the fact that the parameter of the binomial law is the probability of the event p, and therefore it is necessary to indicate from what total number of cases n the probability p(x) is determined. In Poisson's law, the parameter is the average number of cases λ over the period under consideration, so the duration of the period is not directly included in the formula.

Variance of Poisson distribution and third central point are equal to the mathematical expectation, that is, they are also equal to λ.

If there are large differences between the mean and variance, Poisson's law cannot be used. The Poisson distribution is tabulated and given in all collections of statistical tables, reference books and textbooks on statistics. In Fig. Figure 5.2 shows the distribution of the number of days with thunderstorms (a rare event) according to Poisson’s law. For Arkhangelsk for the year λ, = 11 days and for July λ = 4 days. As can be seen from Fig. 5.2, in Arkhangelsk the probability of eight days with a thunderstorm in July is approximately 0.03, and the probability of eight days a year is about 0.10. Let us pay attention to one circumstance. Often, the average number of days with a phenomenon in a year λ for λ≤1 is interpreted as the reciprocal of the repetition period T (for example, λ = 0.3 - one day every three years, λ = 1 - almost annually).

This “averaged” approach is fraught with errors, the greater the larger λ. Even if the days with the phenomenon are not related to each other, years with not one, but several days are likely. As a result, the relation T = 1/λ turns out to be incorrect. Thus, with λ = 1, the phenomenon, as can be easily seen from the formula of Poisson’s law, is observed not annually, but only in 6-7 years out of 10. The probability that the phenomenon will not be observed in a year is equal to the probability that there will be one day with the phenomenon (0.37) and almost the same as the probability that there will be two or more days. Only at λ≤ 0.2 can the indicated relationship be used with sufficient justification; because the probability of two or more days a year in this case is less than 0.02 (less than once every 50 years).

The application of Poisson's law to rare meteorological events is not always useful. For example, sometimes rare phenomena can follow one another due to the fact that the conditions that cause them persist long time, and the conditions of Poisson's law are not satisfied.

More in keeping with the nature of rare meteorological phenomena complex Poisson distribution (negative binomial distribution). It arises when a number of phenomena can be considered as values ​​of different random variables (samples from different populations). All these quantities have a Poisson distribution, but with different parametersλ 1, λ 2 ..., λ k.

The complex Poisson distribution depends, on the one hand, on the distribution of the set of parameters, and on the other, on the distribution of each of the values. Expression for the probability in case given distribution looks like

(5.2)

or in a form more convenient for calculations

The mathematical expectation M and variance D of this distribution are related to its parameters γ and λ by the formulas

(5.3)

Replacing the values ​​of M and D with their estimates and , we obtain

(5.4)

Calculations p(x) can be simplified by taking advantage of the fact that there is equality

, (5.5)

. (5.6)

Hence,

Calculation example. Let's calculate the distribution of the number of days with strong wind at the station Chulym for July, if =1 day, σ=1.7 days. Let us define α and γ:

α≈

γ≈

The probability of not having a single day with strong winds is

p(0)=

The probability that there will be one day with strong wind is p(1)= . The graph of the complex Poisson distribution is shown in Fig. 5.3.

For continuous random variables in climatology, the most commonly used distributions are normal, lognormal, Charlier distribution, gamma distribution, Weibull and Gumbel distributions, as well as the composition law of normal and uniform density.

The greatest theoretical and practical significance has a normal, or Gaussian, distribution law. This law is the limit for many others theoretical distributions and is formed when each value of a random variable can be considered as a sum of sufficiently large number independent random variables.

The normal law is given by expressions for the density and distribution function of the form

When considering the basic principles of probability theory and mathematical statistics and determining distribution parameters, we proceeded from the assumption that a sufficiently large infinite number tests n®N (N®¥), which is practically impossible to implement.

However, there are methods that allow you to estimate these parameters from a sample (part) random events.

General is the set of all conceivable values ​​of observations that we could make under a given set of conditions. In other words, all possible realizations of a random variable, theoretically in the limit there can be an infinite number of them (N®¥). Part of this totality nÎN, i.e. the results of a limited series of observations x 1 , x 2 ,..., x n of a random variable can be considered as a sample value of a random variable (for example, when determining the chemical composition of alloys, their mechanical strength, etc.). If all the ingots of a given grade of steel, cast iron, alloy are cut into samples and examined chemical composition, mechanical strength and others physical characteristics, then they would have a general population of observations. In fact, it is possible (expedient) to study the properties of a very limited number of samples - this is their sampling population.

Based on the results of such a limited number of observations, it is possible to determine point estimates distribution laws and their parameters. An estimate (or sample statistic) Q* of some parameter Q is called arbitrary function Q*=Q*(x 1, x 2,..., x n) of observed values ​​x 1, x 2,..., x n, reflecting to one degree or another real value parameter Q.

If we talk about the characteristics of probability distributions, then the characteristics of theoretical distributions (M x, s x 2, M o, M e) can be considered as characteristics existing in the general population, and characterizing empirical distribution– as their selective characteristics (assessments). Numerical parameters for estimating M x, s x 2, etc. are sometimes called statistics.

For rate mathematical expectation the arithmetic mean (average value) of a number of measurements in the sample is used:

where x i is the implementation of either a discrete or a separate point for a continuous random variable; n – sample size.

To characterize the spread of a random variable, an estimate of the theoretical variance is used - sample variances (see Fig. 2.4):

(3.2a)

(3.2b)

Non-negative value square root from the sample variance is the sample standard deviation(sample standard) deviation

It should be noted that in any problem involving measurement, there are two possible ways to obtain an estimate of the value of s x 2.

When using the first method, a sequence of instrument readings is taken and by comparing the results obtained with a known or calibrated value of the measured quantity, a sequence of deviations is found. The resulting sequence of deviations is then used to calculate the average square deviation according to formula (3.3a).

The second way to obtain an estimate of the value of s x 2 is to determine the arithmetic mean, since in this case, the actual (exact) value of the measured quantity is unknown. In this case, it is advisable to use another formula to find standard deviation(3.2b, 3.3b). Division by (n-1) is done because best estimate, obtained by averaging the X array, will differ from exact value by some amount if a sample is considered rather than the entire population.

In this case, the sum of squared deviations will be slightly less than when using the true average . Dividing by (n-1) instead of n will partially correct this error. In some manuals mathematical statistics It is recommended to always divide by when calculating the sample standard deviation, although sometimes this should not be done. It is necessary to divide by only in cases where the true value has not been obtained by an independent method.

The sample value of the coefficient of variation n, which is a measure of the relative variability of the random variable, is calculated using the formula

or as a percentage

(3.4b)

The one of the samples has a larger scattering and the variation is greater.

Estimations , S x 2 are subject to the requirements of consistency, unbiasedness and efficiency.

An estimate of the parameter Q* is said to be consistent if, as the number of observations n grows (i.e., n®N in the case of a finite population of volume N and with n®¥ in the case of an infinite population), it tends to the estimated theoretical value of the parameter

For example, for variance

(3.5)

An estimate of the parameter Q* is called unbiased if its mathematical expectation M(Q*) for any n asymptotically tends to the true value M(Q*)=Q. Satisfying the requirement of unbiasedness eliminates the systematic error in parameter estimation, which depends on the sample size n and, if consistent, tends to zero at n®¥. Above, two estimates were defined for the variance and . When unknown value mathematical expectation (true value of the measured quantity), both estimates are consistent, but only the second (3.2b), (3.3b), as was shown earlier, is unbiased. The requirement of unbiasedness is especially important with a small number of observations, since when n®¥ ® .

An estimate of the parameter Q 1 * is called effective if, among other estimates of the same parameter Q 2 *, Q 3 *, it has the smallest variance.

(3.6)

where Q i * is any other estimate.

So, if there is a sample x 1, x 2,..., x n from the general population, then the average mathematical expectation can be estimated in two ways:

(3.7)

where x max (n), x min (n) – respectively maximum and minimum value random variable from sample n.

Both estimates have the properties of consistency and unbiasedness, however, it can be shown that the variance in the first method of estimation is equal to S x 2 /n, and in the second, p 2 S x 2 /, i.e. significantly more. Thus, the first method of estimating the mathematical expectation is consistent, unbiased and effective, and the second is only consistent and unbiased. Note that of all unbiased and consistent estimates, one should prefer the one that turns out to be closest to the estimated parameter.

Note that all of the above applies to equal-precision measurements, i.e. to measurements that contain only a random error subject to normal law distributions.

Variation series. Polygon and histogram.

Distribution range- represents an ordered distribution of units of the population being studied into groups according to a certain varying characteristic.

Depending on the characteristic underlying the formation of the distribution series, they are distinguished attributive and variational distribution rows:

§ Distribution series constructed in ascending or descending order of values quantitative characteristic are called variational.

The variation series of the distribution consists of two columns:

The first column contains quantitative values variable trait, which are called options and are designated . Discrete option - expressed as an integer. The interval option ranges from and to. Depending on the type, options can be constructed discrete or interval variation series.
The second column contains number of specific option, expressed in terms of frequencies or frequencies:

Frequencies- This absolute numbers, showing the number of times cumulatively occurs given value signs that denote . The sum of all frequencies must be equal to the number of units in the entire population.

Frequencies() are frequencies expressed as a percentage of the total. The sum of all frequencies expressed as percentages must be equal to 100% in fractions of one.

Graphic image distribution series

The distribution series are visually presented using graphical images.

The distribution series are depicted as:

§ Polygon

§ Histograms

§ Cumulates

Polygon

When constructing a polygon, the values ​​of the varying characteristic are plotted on the horizontal axis (abscissa axis), and on vertical axis(y-axis) - frequencies or frequencies.

1. Polygon in Fig. 6.1 is based on data from the micro-census of the population of Russia in 1994.

bar chart

To construct a histogram, the values ​​of the boundaries of the intervals are indicated along the abscissa axis and, based on them, rectangles are constructed, the height of which is proportional to the frequencies (or frequencies).

In Fig. 6.2. shows a histogram of the distribution of the population of Russia in 1997 by age groups.

Fig.1. Distribution of the Russian population by age groups

Empirical function distributions, properties.

Let it be known statistical distribution frequencies of a quantitative characteristic X. Let us denote by the number of observations in which a value of the characteristic was observed that was less than x and by n – total number observations. Obviously, the relative frequency of event X

An empirical distribution function (sampling distribution function) is a function that determines for each value x the relative frequency of the event X

In contrast to the empirical distribution function of a sample, the population distribution function is called the theoretical distribution function. The difference between these functions is that the theoretical function determines the probability of event X

As n increases, the relative frequency of the event X

Basic properties

Let an elementary outcome be fixed. Then is the distribution function of the discrete distribution given by the following probability function:

where and - number of sample elements equal to . In particular, if all elements of the sample are different, then .

The mathematical expectation of this distribution is:

.

Thus, the sample mean is the theoretical mean of the sampling distribution.

Similarly, sample variance is the theoretical variance of a sampling distribution.

The random variable has a binomial distribution:

The sample distribution function is an unbiased estimate of the distribution function:

.

The variance of the sample distribution function has the form:

.

According to the strong law of large numbers, the sample distribution function converges almost certainly to the theoretical distribution function:

almost certainly at .

The sample distribution function is an asymptotically normal estimate of the theoretical distribution function. If , then

According to the distribution at .

Empirical distribution function

ED processing methods are based on the basic concepts of probability theory and mathematical statistics. These include the concepts of general population, sample, empirical distribution function.

Under general population understand all possible parameter values ​​that can be recorded during an unlimited time observation of an object. Such a set consists of an infinite number of elements. As a result of observing an object, a limited-in-volume set of parameter values ​​is formed x 1 , x 2 , …, xn. From a formal point of view, such data represent sample from the general population.

We will assume that the sample contains complete developments before system events (there is no censoring). Observed values x i called options , and their number is sample size n. In order for any conclusions to be drawn from the observation results, the sample must be representative(representative), i.e. correctly represent the proportions of the general population. This requirement is met if the sample size is large enough and each element in the population has the same probability of being included in the sample.

Let the resulting sample have a value x 1 parameter observed n 1 time, value x 2 – n 2 times, meaning xk – nk once, n 1 +n 2 + … +nk=n.

A set of values ​​written in ascending order is called variation series, quantities n i – frequencies, and their relationship to the sample size ni=n i /n – relative frequencies(frequencies). Obviously, the sum of the relative frequencies is equal to unity.

Distribution refers to the correspondence between observed variants and their frequencies or frequencies. Let nx – the number of observations in which the random values ​​of the parameter X less x. Event Frequency X equal to nx/n. This ratio is a function of x and on sample size: F n(x)=nx/n. Magnitude Fn(x) has all the properties of a function:

distributions: Fn(x) non-decreasing function, its values ​​belong to the segment ;

If x 1 is the smallest value of the parameter, and xk – the greatest, then Fn(x)= 0, When x<x 1 , And FP(xk)= 1 when x>=xk.

Function Fn(x) is determined by ED, which is why it is called empirical distribution function. Unlike the empirical function Fn(x) distribution function F (x) of the population is called the theoretical distribution function, it characterizes not the frequency, but the probability of an event X<x. From Bernoulli's theorem it follows that the frequency Fn(x) tends in probability to probability F(x) with unlimited magnification n. Consequently, with a large volume of observations, the theoretical distribution function F(x) can be replaced by the empirical function Fn(x).

Graph of empirical function Fn(x) is a broken line. In the spaces between adjacent members of the variation series Fn(x) remains constant. When passing through axis points x, equal to the sample members, Fn(x) undergoes a discontinuity, increasing abruptly by the value 1/ n, and if there is a coincidence l observations - on l/n.

Example 2.1. Construct a variation series and graph of the empirical distribution function based on the observation results, table. 2.1.

Table 2.1

The desired empirical function, Fig. 2.1:

Rice. 2.1. Empirical distribution function

With a large sample size (the concept of “large volume” depends on the goals and processing methods, in this case we will consider P big if n>40) for the convenience of processing and storing information resort to grouping EDs into intervals. The number of intervals should be chosen so that the variety of parameter values ​​in the aggregate is reflected to the required extent and at the same time the distribution pattern is not distorted by random frequency fluctuations in individual categories. There are loose guidelines for choosing quantities y And size h such intervals, in particular:

each interval must contain at least 5–7 elements. In extreme ranks, only two elements are allowed;

the number of intervals should not be very large or very small. Minimum the y value must be at least 6 – 7. With a sample size not exceeding several hundred elements, the value y is set in the range from 10 to 20. For a very large sample size ( n>1000) the number of intervals may exceed the specified values. Some researchers recommend using the ratio y=1.441*ln( n)+1;

with relatively small unevenness in the length of the intervals, it is convenient to choose the same and equal to the value

h= (x max – x min)/y,

Where x max – maximum and x min – minimum value of the parameter. If the distribution law is significantly uneven, the length of the intervals can be set to a smaller size in the region of rapid changes in the distribution density;

If there is significant unevenness, it is better to assign approximately the same number of sample elements to each category. Then the length of a particular interval will be determined by the extreme values ​​of the sample elements grouped into this interval, i.e. will be different for different intervals (in this case, when constructing a histogram, normalization by the length of the interval is required - otherwise the height of each element of the histogram will be the same).

Grouping observation results by intervals provides for: determining the range of changes in a parameter X; choosing the number of intervals and their size; counting for everyone i- th interval [ xi–xi+1 ] frequencies ni or relative frequency (frequency n i) options fall into the interval. As a result, a representation of the ED is formed in the form interval or statistical series.

Graphically, a statistical series is displayed in the form of a histogram, polygon and stepped line. Often histogram represented as a figure consisting of rectangles, the bases of which are intervals of length h, and the heights are equal to the corresponding frequency. However, this approach is inaccurate. Height i- th rectangle z i should be chosen equal ni/ (nh). Such a histogram can be interpreted as a graphical representation of the empirical distribution function fn(x), in it the total area of ​​all rectangles will be one. The histogram helps to select the type of theoretical distribution function for approximating the ED.

Polygon called a broken line, the segments of which connect points with coordinates along the abscissa axis equal to the midpoints of the intervals, and along the ordinate axis equal to the corresponding frequencies. The empirical distribution function is displayed as a stepped broken line: a horizontal line segment is drawn over each interval at a height proportional to the accumulated frequency in the current interval. The accumulated frequency is equal to the sum of all frequencies, starting from the first and up to this interval inclusive.

Example 2.2. There are results of recording signal attenuation values xi at a frequency of 1000 Hz of the switched channel of the telephone network. These values, measured in dB, are presented in the form of a variation series in table. 2.3. It is necessary to construct a statistical series.

Table 2.3

i
xi	25,79	25,98	25,98	26,12	26,13	26,49	26,52	26,60	26,66	26,69	26,74
i
xi	26,85	26,90	26,91	26,96	27,02	27,11	27,19	27,21	27,28	27,30	27,38
i
xi	27,40	27,49	27,64	27,66	27,71	27,78	27,89	27,89	28,01	28,10	28,11
i
xi	28,37	28,38	28,50	28,63	28,67	28,90	28,99	28,99	29,03	29,12	29,28

Solution. The number of digits of the statistical series should be chosen as minimal as possible to ensure a sufficient number of hits in each of them; let’s take y = 6. Let’s determine the size of the digit

h =(x max – x min)/y =(29.28 – 25.79)/6 = 0.58.

Let's group observations by category, table. 2.4.

Table 2.4

i
xi	25,79	26,37	26,95	27,5 3	28,12	28,70
ni
n i=ni/n	0,114	0,205	0,227	0,205	0,11 4	0,136
z i =n i /h	0,196	0,353	0,392	0,353	0,196	0,235

Based on the statistical series, we will construct a histogram, Fig. 2.2, and the graph of the empirical distribution function, Fig. 2.3.

Graph of the empirical distribution function, Fig. 2.3 differs from the graph presented in Fig. 2.1 equality of the change step of the options and the size of the increment step of the function (when constructed using a variation series, the increment step is a multiple

1/ n, and according to the statistical series - depends on the frequency in a particular category).

The considered ED representations are the initial ones for subsequent processing and calculation of various parameters.

Lecture 13. The concept of statistical estimates of random variables

Let the statistical frequency distribution of a quantitative characteristic X be known. Let us denote by the number of observations in which the value of the characteristic was observed to be less than x and by n the total number of observations. Obviously, the relative frequency of event X< x равна и является функцией x. Так как эта функция находится эмпирическим (опытным) путем, то ее называют эмпирической.

Empirical distribution function(sampling distribution function) is a function that determines for each value x the relative frequency of the event X< x. Таким образом, по определению ,где - число вариант, меньших x, n – объем выборки.

In contrast to the empirical distribution function of a sample, the population distribution function is called theoretical distribution function. The difference between these functions is that the theoretical function determines probability events X< x, тогда как эмпирическая – relative frequency the same event.

As n increases, the relative frequency of the event X< x, т.е. стремится по вероятности к вероятности этого события. Иными словами

Properties of the empirical distribution function:

1) The values ​​of the empirical function belong to the segment

2) - non-decreasing function

3) If is the smallest option, then = 0 for , if is the largest option, then = 1 for .

The empirical distribution function of the sample serves to estimate the theoretical distribution function of the population.

Example. Let's construct an empirical function based on the sample distribution:

Options
Frequencies

Let's find the sample size: 12+18+30=60. The smallest option is 2, so =0 for x £ 2. The value of x<6, т.е. , наблюдалось 12 раз, следовательно, =12/60=0,2 при 2< x £6. Аналогично, значения X < 10, т.е. и наблюдались 12+18=30 раз, поэтому =30/60 =0,5 при 6< x £10. Так как x=10 – наибольшая варианта, то =1 при x>10. Thus, the desired empirical function has the form:

The most important properties of statistical estimates

Let it be necessary to study some quantitative characteristic of the general population. Let us assume that from theoretical considerations it has been possible to establish that which one exactly the distribution has a sign and it is necessary to evaluate the parameters by which it is determined. For example, if the characteristic being studied is distributed normally in the population, then it is necessary to estimate the mathematical expectation and standard deviation; if the characteristic has a Poisson distribution, then it is necessary to estimate the parameter l.

Typically, only sample data is available, for example, values ​​of a quantitative characteristic obtained as a result of n independent observations. Treated as independent random variables we can say that to find a statistical estimate of an unknown parameter of a theoretical distribution means to find a function of observed random variables that gives an approximate value of the estimated parameter. For example, to estimate the mathematical expectation of a normal distribution, the role of the function is played by the arithmetic mean

In order for statistical estimates to provide correct approximations of the estimated parameters, they must satisfy certain requirements, among which the most important are the requirements undisplaced And solvency assessments.

Let be a statistical estimate of the unknown parameter of the theoretical distribution. Let the estimate be found from a sample of size n. Let's repeat the experiment, i.e. let's extract another sample of the same size from the general population and, based on its data, obtain a different estimate. Repeating the experiment many times, we get different numbers. The score can be thought of as a random variable, and the numbers as its possible values.

If the estimate gives an approximate value in abundance, i.e. each number is greater than the true value, and as a consequence, the mathematical expectation (average value) of the random variable is greater than:. Likewise, if it gives an estimate with a disadvantage, That .

Thus, the use of a statistical estimate, the mathematical expectation of which is not equal to the estimated parameter, would lead to systematic (of the same sign) errors. If, on the contrary, then this guarantees against systematic errors.

Unbiased called a statistical estimate, the mathematical expectation of which is equal to the estimated parameter for any sample size.

Displaced is called an estimate that does not satisfy this condition.

The unbiasedness of the estimate does not yet guarantee a good approximation for the estimated parameter, since possible values ​​can be very scattered around its average value, i.e. the variance can be significant. In this case, the estimate found from the data of one sample, for example, may turn out to be significantly distant from the average value, and therefore from the parameter being estimated.

Effective is a statistical estimate that, for a given sample size n, has smallest possible variance .

When considering large samples, statistical estimates are required to solvency .

Wealthy is called a statistical estimate, which, as n®¥ tends in probability to the estimated parameter. For example, if the variance of an unbiased estimate tends to zero as n®¥, then such an estimate turns out to be consistent.