What is the secret of the best sellers? Just watch out for the best sellers, be it a company, remember that they are one hell of a thing. The skin of them works with a great number of people to make more presentations, less successful salespeople. Cі people rozumіyut, scho sales - count of numbers, and the more people stink tell about their product services, they lay more pleasing - the axis and everything. If you think about it, that it’s not enough to talk to them not enough, who will say “so” to them in a different way, but to those whose interest to their proposition is not so great, then the law of averages is right for their greed.

Your income will depend on the number of sales, but at the same time the stink will be directly proportional to the number of presentations, as you work. As soon as you understand it, you will begin to put into practice the law of averages, anxiety, connected with the beginning of a new business, work in a new sphere, will be lowered. And as a result, it is more likely to be more controllable than the level of earning in one's building. If you are just making a presentation and being aware of your newcomers in the process, please show up.

Why think about the number of things, think more about the number of presentations. There is no sense to wander around or come home in the evening and guess who will buy your product. Natomist, plan your day more beautifully, you need to work more than a few dzvinks. And then, no matter what, ring out all the rings! Such a pidhid to ask you a robot - because that specific meta is simple. If you know that you have to stand in front of you all the time, it will be easier for you to make the planned number of calls. If in this process you once feel "so" - it's better!

And if “nі”, then in the evening you will see that you honestly killed everything you could, and you won’t be tormented by thoughts about those, how many pennies you earned, otherwise you’ve gained a lot of companions in a day.

For example, in your company or in your business, the average salesperson puts one favor on some presentation. Now show yourself that you are drawing cards from the deck. A skin card of three suits - spades, tambourines and clubs - this is a presentation, in which case you professionally represent a product, in the service of opportunity. Do the best you can, as little as you can, but all the same you can’t please. And the skin worm's card is a favor that allows you to take pennies or get a new companion.

In such a situation, why don't you want to draw more cards from the deck? Let's say you win the card styles, you want the skill cards, and if you want to pay you, or you should show the new companion to the schoroza, if you win the card of hearts. You will remember how to pull the cards, the ice is pomіchayuchi, which suits the card smartly.

You know that a deck of fifty two cards has thirteen of hearts. And two decks have twenty six heart cards, and so on. Chi will you rozcharovani, vityagnuvshi write-offs, tambourines chi clubs? No great! Do you think only about those that such a skin "miss" is approaching you - to what? To the heart's card!

But you know what? You have already been told such a proposition. You are in a unique situation that allows you to earn skills, you want to win skills, and you want to win skills in your life. And if you just summarily "pull the cards", improve your skills and endure three peaks, bubo and clubs, then you will become a miracle seller and achieve success.

One of the speeches to slow down the process of selling the flooring to the greedy is those who once shuffle the deck, the cards will be mixed differently. Every once in a while all the worms appear on the cob of the deck, and then the smoky distance (if we are already in trouble, we are by no means programmed!) We are checked for a long row of cards of a different suit. And suddenly, for the first worms to get away, it happens to pass through an inexhaustible number of peaks, a club and a bubo. And sometimes cards of different suits fall out of thin air. Ale, be it in a flurry, in a leather deck of fifty two cards, in any order, always have thirteen heart cards. Just take the cards doti, you don't know the doki.

Type: Leylya,  

Yakshcho a manifestation of resilience middle if it is true, then in mathematical models, for the help of which we are developing vipadkovі phenomena, is guilty of establishing a theorem that reflects this fact.
At the time of the theorem, we introduce a substitution on the reverse values X 1 , X 2 , …, X n:

a) cutaneous vipadical value Х i maє mathematical ochіkuvannya

M(Х i) = a;

b) skin dispersion vipadkovy size kіntseva, otherwise, we can say that dispersions are surrounded by the beast with one and the very same number, for example Z, then.

D(Х i) < C, i = 1, 2, …, n;

c) variable values ​​are pairwise independent, so be two X iі Xj at i¹ j independent.

Todd, obviously

D(X 1 + X 2 + … + X n)=D(X 1) + D(X 2) + ... + D(X n).

We formulate the law of great numbers in Chebishev's words.

Chebishev's theorem: with an unrestricted increase in numbers n independent testing " the arithmetic mean is the value of the variable value, which they fear to converge in terms of similarity to the її mathematical ochіkuvannya. ”, then for some kind of positive ε

R(| –a| < ε ) = 1. (4.1.1)

sense expression "Arithmetic mean = converge smoothly up to a " believe in what is imovirnist of what skіlki will always be little vіdіznyatis vіd a, unobtrusively approaching up to 1st increase n.

Bringing. For the last number n independent testing of Chebishev’s unevenness for the vertical value = :

R(|-M()| < ε ) ≥ 1 – . (4.1.2)

Vrahovoyuchi exchange a - b, calculable M( ) that D( ):

M( ) = = = = = = a;

D( ) = = = = = = .

Substituting M( ) that D( ) at unevenness (4.1.2), we take

R(| –a| < ε )≥1 – .

Even in unevenness (4.1.2) take skilki invariably small ε >0і n® ¥, then we take

how to prove Chebishev's theorem.

From the examined theorem, there is an important practical vysnovok: we can’t replace the mathematical value of the vipadkovy value with the arithmetic mean values, we’ll take it for a great number of additions. In case of tsomu, the more doslidіv for calculation, tim with more imovirnіst (nadіynіstyu) it will be cleared up, that a pardon is due to the replacement ( - a) do not exceed the given value ε .

From the other side, you can virishuvati іnshi practical tasks. For example, for the meaning of imovirnosti (nadіynostі) R=R(| – a|< ε ) and maximum allowable pardon ε indicate the necessary number of doslidiv n; on Rі P signify ε; on ε і P choose between imovirnosti podії | – a |< ε.

Okremy vipadok. Come on at n try to beware n slope value x, maє mathematical ochіkuvannya M(X) and dispersion D(X). Otrimanі value is possible as a variable value X 1 ,X 2 ,X 3 , ... ,X n,. Tse next to understand this: series z P testing is carried out more than once, so as a result i-th test, i= l, 2, 3, ..., P, in the skin series of tests, those chi appear lower than the value of the depression value X, do not see the distance. Otzhe, i-e value x i vipadkovy size, otrimane in i-m trials are changed by a vipadkovy rank, so you go to one series of trials to another. In such a rank, skin meaning x i can be entered by the vipad value Xi.

It is acceptable that the test is to please the advancing vimog:

1. Testing independent. Tse means what results X 1 , X 2 ,
X 3 , ..., X n viprobuvan - independent vipadkovі values.

2. Testing is carried out in the same minds - it means, from the point of view of the theory of mindfulness, that the skin of vipadkovyh values X 1 ,X 2 ,X 3 , ... ,X n May such a law itself have been subdivided, that the magnitude of X to that M(X i) =M(X)і D(X i) = D(X), i = 1, 2, .... P.

Mind you, take it away

R(| –a| < ε )≥1 – . (4.1.3)

Example 4.1.1. X 4. How much do you need to work out the independent numbers, if it is not less than 0.9 could be scored, what is the arithmetic mean of the value of the drop value to be considered as a mathematical score less than 0, 5?

Solution. Behind the mind ε = 0,5; R(| – a|< 0,5) ≥ 0.9. Using the formula (4.1.3) for the vipadical quantity X, taken

P(|-M(X)| < ε ) ≥ 1 – .

Zі spіvіdnoshennia

1 – = 0,9

significant

P= = = 160.

Vidpovid: you need 160 independent doslidiv.

Just let it go, that the arithmetic mean rozpodіlena normally, then otrimuemo:

R(| – a|< ε )= 2Φ () ≥ 0,9.

The sounds, shortened by the table of the Laplace function, are taken ≥
≥ 1.645, or 6.58, tobto. n ≥49.

Example 4.1.2. Dispersion of the fall value X goodbye D( X)=5. . Deputy of the unknown meaning of mathematical honing a accepted . Designate the maximum value of the pardon, which is allowed with a minimum of 0.8.

Solution. Behind the mind n= 100, R(| –a|< ε ) ≥0.8. Let us solve the formula (4.1.3)

R(| –a|< ε ) ≥1 – .

Zі spіvіdnoshennia

1 – = 0,8

significant ε :

ε 2 = = = 0,25.

Otzhe, ε = 0,5.

Vidpovid: maximum pardon value ε = 0,5.

4.2. The law of great numbers in Bernoulli's form

Wanting to be based on the understanding of imovirnosti, whether in the basis of some kind of statistical visnovka, we can only in some cases indicate imovirnіst podії without intermediary. Differentiation can be installed from the world of symmetry, equal possibility, but only to a universal method, which allows bi for a sufficient degree to show it, it does not. Bernoulli's theorem gives the possibility of an approximate assessment of imovirnosti, as if for a podії, scho to tell us. BUT you can conduct repeated independent testing. Let it be broken P independent samples BUT fast and equal nar.

Bernoulli's theorem. With uncircumcised growth, the number of independent samplings P visible frequency BUT converge immovably to imovirnosti p appear podії BUT,T. e.

P(½ - p½≤ ε) = 1, (4.2.1)

de ε - skіlki zavgodno small positive number.

For the final n for mind, scho, Chebishev's unsteadiness for the vipadkovy value of matime looked:

P(| - p |< ε ) ≥ 1 – .(4.2.2)

Bringing. We prove Chebishev's theorem. Come on X i- Number of appearances BUT in i-om tested, i= 1, 2, . . . , n. Skin size X i You can take only two values:

X i= 1 (pod_ya BUT it has come) with imovirnistyu p,

X i= 0 (podіya BUT not come) with imovirnistyu q= 1- p.

Come on Y n=. Suma X 1 + X 2 + … + X n up to the number m appear podії BUT in n testing (0 m n), which means Y n= - Visible frequency of appearance BUT in n samples. Mathematically refined and variance X i equal:

M( ) = 1∙p + 0∙q = p,

Stock 4.2.1. With the method of installing a part of the product line, 1000 units were re-examined according to the scheme of the round-robin. How is the possibility of what is installed in the number of samples of the slub for the absolute value of the slub in the entire batch no more than 0.01 lower, as in the average on the skin 10,000 defects fall 500?

Solution. For the mental task of a number of independent tests n= 1000;

p= = 0,05; q= 1 – p= 0,95; ε = 0,01.

Zastosovuyuchi formula (4.2.2), we take

P(| –p|< 0,01) ≥ 1 – = 1 – = 0,527.

Vidpovid: with a change of not less than 0.527 it should be considered that the vibrating part of the gateway (the frequency of the appearance of the gateway) is charged with a part of the gateway for all products (in the case of the quality of the gateway) no more than 0.01.

Stock 4.2.2. When stamping parts, the sluice can become 0.05. How many details are needed to check the details, so that it is not less than 0.95?

Solution. Behind the mind R= 0,05; q= 0,95; ε = 0,01;

P(| – p|<0,01) ≥ 0,95.

Rivnosti 1 – = 0.95 known n:

n= = =9500.

Vidpovid: 9500 parts must be checked.

Respect. Estimates of the necessary number of warnings, as they win when stosuvanni Bernoulli's (abo Chebishev's) theorems, are even more. Establish more precise estimates, suggested by Bernstein and Khinchin, but require a sophisticated mathematical apparatus. In order to avoid rebіlshennya estimates, іnоdі koristuyutsya Laplace's formula

P(| – p|< ε ) ≈ 2Φ .

A small amount of the formula is the full extent of the assessment of the error, which is allowed.

LECTURE 5

Repetition of the past

Part 1 - ROZDIL 9. THE LAW OF GREAT NUMBERS. Boundary THEOREMS

For statistical appointment
Imovirnosti won't take it like a deak
number
The frequency of the vipadkovy under. At
axiomatically assigned imovirnist –
ce, in fact, additive multiplier
results
podії. The first view may be on the right
empirical boundary, the other - s
theoretical concepts of the world. Zovsim NOT
it is obvious that they stink to one and the same
understand. Call of the clergy
Imovirnosti establish Bernoulli's theorem,
sho є we will adorn with vipadka the law of the great
numbers.

With an increase in the number of samples
binomial law of pragne
normal distribution. Tse theorem
Moivre-Laplace, yak є
let's call the depression of the central boundary
theorems. It remains to say what the function
rozpodіlu sumi independent
variable values ​​from increasing numbers
dodankiv pragne normal
law.
The law of great numbers is central
boundary theorem lie at the base
mathematical statistics.

9.1. Chebishev's nervousness

Let the vipadic value ξ maє
final mathematical education
M[ξ] and variance D[ξ]. Same for
any positive number ε
fair inconsistency:

Notes

For protilezhny podії:
Chebishev's unevenness is valid for
let me follow some law.
poklavshi
fact:
, at least non-trivial

9.2. The law of great numbers in Chebishev's form

Theorem
pairwise independent and mayut kіntsevi
dispersions, exchanges
fast
Same for
whatever
maybe
In this rank, the law of great numbers talk about
zbіzhnostі for ymovіrnіstyu arithmetic mean vypadkovy values ​​(tobto vypadkovy value)
to the arithmetic mean їх mat. ochikuvan (tobto.
up to non-vipadkovo size).

9.2. The law of great numbers in the form of Chebishev: additions

Theorem (Markov): the law of the great
numbers are counted, as is the variance
sum of vipadic values ​​does not increase
a little too fast zі rostannyam n:

10.9.3. Bernoulli's theorem

Let's look at Bernoulli's scheme.
Come on μn - the number of come down A in
n independent trials, p - the ability of the current test A in one
testing. Todi for be-whom
Tobto. imovirnіst that, scho vіdhilennya
outward frequency
yoga ymovіrnosti r will be modulo yak zavgodno
few
try n.

11.

Proof: Vipadian value μn
rozpodіlena for the binomial law, that
maybe

12.9.4. Characteristic functions

Characteristic vipadical function
quantity is called a function
de exp(x) = ex.
in such a manner,
is himself
mathematical refinement
complex slope quantity
pov'yazanoї іz size. Zokrema, yakcho
is a discrete vipad value,
the order of subdivision (xi, pi) is given, de i
= 1, 2,..., n, then

13.

For an uninterrupted drop-down quantity
іz shіlnіstyu rozpodіlu
imovirnosti

14.

15.9.5. Central boundary theorem (Lyapunov theorem)

16.

Repeated

17. FOUNDATIONS OF THE THEORY OF POSSIBILITIES AND MATHEMATICAL STATISTICS

PART II. MATHEMATICAL
STATISTICS

18. Epigraph

“I see three kinds of nonsense: nonsense,
damn bullshit those statistics "
Benjamin Disraeli

19. Entry

Two main tasks of mathematical
statistics:
selection and grouping of statistical
danih;
development of methods for analysis
taking away data in the fallow land
goals of follow-up.

20. Methods of statistical analysis of data:

assessment of unknown immovability;
assessment of unknown function
rozpodіlu;
assessment of the parameters of the house
rozpodіlu;
re-verification of statistical hypotheses about the species
unknown rozpodіlu abo about
values ​​of the parameters
rozpodіlu.

21. DIVISION 1. BASIC CONCEPTS OF MATHEMATICAL STATISTICS

22.1.1. General marriage and selection

General sukupnіst - everything
impersonal additional objects,
Vibіrka – collection of objects vipadkovo
selected from the general marriage
for follow up.
Obligation of general marriage
obsyag vibirki - the number of objects in the general marriage and the selection - will be
designate as N and n.

23.

Vibirka will be repeated, if
skin selection object before
by choosing an offensive turn up to
general suupnist, that
without repetition, as if vibrating
the object of the general purchase is not
turn around.

24. Representative selection:

correctly represent the features
general marriage, tobto. є
representative (representative).
Behind the law of great numbers, you can harden
scho tsya umova vykonuєtsya, yakscho:
1) obsyag vibirki n dosit great;
2) the skin of the object of the vibrating was selected in a vipadkovo way;
3) for the skin object, the possibility of drinking
the vibrator has the same.

25.

General marriage and selection
can be the same
(single factor)
and rich-to-factory

26.1.2. The vibratory law has been subdivided (statistical series)

Let the vibrators obsyagom n
tsіkava for us vipadkovy value ξ
(which object parameter
general marriage) accept n1
times x1, n2 times x2,... i
nk times – xk value. Todi are afraid
values ​​x1, x2,..., xk
ξ are called variants, and n1, n2,..., nk
- xx frequencies.

27.

Retail xmax - xmin є range
vibirki, vіdnoshennia ωi = ni /n -
visible frequency of variant xi.
Obviously what

28.

If we write down the options in the order that grows, then we take away the variation row. The table that consists of such
ordering variant of those frequencies
(i/or external frequencies)
is called the statistical series or
vibirkovy law rozpodіlu.
- An analogue of the law of discrete subdivision
vipadkovy value in the theory of imovirnosti

29.

Such a variational series is folded over
great number of numbers
continue without interruption
a sign that vikoristovuyut groupovanu
vibrka. For її otrimannya _interval,
which way everyone is afraid
meaning signs that break into
dekilka sing of equal parts
(pіd_intervalіv) dovzhina h. At
folding the statistical series into
vibrate the middle
pіd_intervalіv, and ni equate to number
variant, which was consumed in the i-th interval.

30.

40
- Frequencies -
35
30
n2
n3
ns
n1
25
20
15
10
5
0
a
a+h/2 a+3h/2
- Options -
b-h/2
b

31.1.3. Polygon of frequencies, selection function

We add the value of the vertical value xi according to
the abscissa axis, and the ni value - along the ordinate axis.
Laman line
points with coordinates (x1, n1), (x2, n2),..., (xk,
nk), is called a polygon
frequencies. Like a deputy
absolute value ni
on the y-axis of the class
visible frequencies ωi,
then we take the polygon of visible frequencies

32.

By analogy with the function of rozpodіlu
discrete drop-down value for
vibirkov law rozpodіlu can be
induce vibirkov (empiric)
rozpodіlu function
de sumovuvannya vykonuetsya on all
frequencies
variant smaller than x. We respect that
empirical function rozpodіlu
to lie down in obsyagu vibirki n.

33.

On the view of the function
found
for the vipadkovy value ξ
way in the results of statistical data processing, help function
rozpodіlu
pov'yazanu s
general sukupnistyu, call
theoretical. (Call the general
the supply of the flooring is great, so
it is impossible to acquire її, tobto.
doslіdzhuvati її less possible
theoretically).

34.

We appreciate that:

35.1.4. The dominance of the empirical function of the rozpodіlu

Step-parts
looking

36.

One more graphic tribute
click on us vibrki є
histogram - step figure,
what is made up of rectangles, the basics of which are intervals
width h, and heights - vіdrіzki zavdovka
ni/h (frequency histogram) or ωi/h
(Histogram of visible frequencies).
In the first moment
area with histograms
selections n,
to another - alone

37. Butt

38. DIVISION 2. NUMBERS AND CHARACTERISTICS OF THE VIBIRKA

39.

Head of Mathematical Statistics –
take away for a clear choice
information about the general
marriage. Numerical characteristics of a representative sample - assessment of relevant characteristics
doslіdzhuvanoї vipadkovoї value,
tied to the general
sukupnistyu.

40.2.1. Vibration average and vibrancy variance, empirical moments

Vibrkov middle is called
arithmetic mean
option at the vibrator
Vybirkove average vikoristovuetsya for
statistical evaluation of mathematical
ochіkuvannya doslіdzhuvanoy vypadkovy value.

41.

Vibration dispersion is called
value, equal
Vibirkov mean square
repentance -

42.

It's easy to show what wins
come spіvvіdnoshennia, better for
variance calculation:

43.

Other characteristics
variation series є:
mode M0 - options, what can
highest frequency, і median me –
variant, what to add variant
a row into two parts, equal to the number
variety.
2, 5, 2, 11, 5, 6, 3, 13, 5 (mode = 5)
2, 2, 3, 5, 5, 5, 6, 11.13 (median = 5)

44.

By analogy with vіdpovіdnimi
theoretical virazes can
induce empirical moments,
stop for statistical
estimates of cob and central
moments
magnitude.

45.

By analogy with moments
theories
Imovirnosti pochatkovy empirichnym
the moment of the order m is the quantity
central empirical moment
order m -

46.2.2. The power of statistical assessments of parameters in the distribution: insufficiency, efficiency, efficiency

2.2. Power of statistical estimates
parametrіv rozpodіlu: nemіschenіst, effektіvnіst, sposobnіst
After otrimannya statistical estimates
parameters
the value of ξ: vibration mean, vibration dispersion, etc., it is necessary to reconsider,
what a stink є kind close
for specific parameters
theoretical distribution of ξ.
We know how to think, how to think for whom
wince.

47.

48.

Statistical score A* is called
undisturbed, as if mathematically
estimating the value of the evaluated parameter
general marriage A for whatever
obsyag vibirki, tobto.
As if the mind is not victorious, assessment
is called misplaced.
The immovability of the assessment is not sufficient
mental approximation of statistical
grades A* to the reference (theoretical) value
estimated parameter A.

49.

Rozkid okremih meaning
mean value M
deposit according to the value of dispersion D.
If the dispersion is large, then the value
known for the data of one vibirki,
can mean
evaluated parameter.
Father, for the hopeful
estimating variance D is guilty
be small. Statistical evaluation
called effective, even though
given obligation of choice n won may
I can find the dispersion.

50.

Before statistical estimates
there is more help
ability. The score is called
possible, as for n → won
pragne imovirnosti before
evaluated parameter. We respect that
unbiased assessment will be
possible, even in n → її
variance pragne 0.

51. 2.3. The dominance of vibratory mean

Please note that the options are x1, x2,..., xn
є values
independent, however, rozpodіlenih vypadkovyh values
,
how can mathematically improve
that dispersion
. Todi
vibirkove middle is possible
look like a vipadka value

52.

Immovability. 3 powers
mathematical refinement of the following
tobto. vibirkove average є
unbiased assessment of the mathematical
ochіkuvannya vipadkovy size.
You can also show the effectiveness
scores from the vibratory average mathematical score (for normal
rozpodіlu)

53.

Helpfulness. Let a - scoring
parameter, but the mathematical
singling out the general marriage
- Dispersion of general marriage
.
Let's look at Chebishev's nervousness
We have:
also
. As n → the right part
unevenness to zero for any ε > 0, then.
i, also, the value of X, which represents vibirkova
estimating, pragne estimating parameter a by imovirnosti.

54.

In this rank, you can sprout visnovok,
what is the average
undisturbed, effective (according to
accept for normal
rozpodіlu) and possible
math score
vipadkovy size, pov'yazanoї z
general sukupnistyu.

55.

56.

LECTURE 6

57. 2.4. The dominance of vibrational dispersion

Vibration dispersion D*
estimating the variance of the fall rate

58.

59.

60. Stock

Know vibirkove average, vibirkove
variance and rms
vіdhilennya, the fashion is corrected vibіrkovu
dispersion for vibrating, what can come
the law was subdivided:
Solution:

61.

62. PARAMETERS

63.

Vvazhatimemo, scho blatant looking at the law
let us know
need to specify details -
parameters, what to designate yoga
design form. Use
sprat of methods
task, two of them
at a glance: the moment method and the method
the greatest credibility

64.3.1. Method of moments

65.

Method of moments, resentments by Karl
Pirson in 1894, foundations on
vikoristannya tsikh nablizhdenih equalities:
moments
get out of insurance
theoretically according to the law
subdivision with the parameters θ, and
vibratory moments
counted
for a clear choice. Unknown
parameters
appointed in
the results of the development of the system r
pov'yazuyut vіdpovіdnі
theoretical and empirical moments,
for example,
.

66.

Can you show what the ratings
parameters θ, taken away by the method
moments, possible, їх
mathematical refinement
view the reference values ​​of the parameters on
value of the order of n-1, and the average
quadratic deviation є
values ​​on the order of n-0.5

67. Butt

Apparently, the characteristics of objects
general marriage, being vipadkovoy
value, maє equal rozpodіl, scho to deposit in the form of parameters a and b:
It is necessary to calculate by the method of moments
parameters a and b according to the vidomim vibrkov
middle
that vibrational dispersion

68. Fortune

α1 - mat.scoring β2 - variance

69.

(*)

70.

71.3.2. Maximum likelihood method

The method is based on the likelihood function
L(x1, x2,..., xn, θ), which is the law
rozpodіl vector
, de
fluctuations in magnitude
accept values
vibirka option, tobto. mayut the same
Rozpodіl. Oskіlki vipadkovі values
independent, the credibility function may look like:

72.

Idea of ​​the largest method
plausibility lies in what we
judging by the same value of the parameters θ, at
some imovirnist appeared in
choose value option x1, x2,..., xn
є the greatest. In other words,
as an estimation of parameters θ
a vector is taken, for which the function
plausibility may be local
maximum for given x1, x2, …, xn:

73.

Estimates for the method of maximum
credibility emerge from
necessary mind the extremum
functions L(x1,x2,..., xn,θ) at the point

74. Notes:

1. For an hour I will search for the maximum function of credibility
for forgiveness of rozrakhunkiv, you can vikonati
dії, that do not change the result: first,
replace L(x1, x2,..., xn,θ) with the logarithmic likelihood function l(x1, x2,..., xn,θ) =
log L(x1, x2,..., xn,θ); in a different way, look at the viraz
for the plausibility function, which does not lie in the form of θ
dodanki (for l) or positive
multiply (for L).
2. Estimates of parameters reviewed by us,
can be called point estimates, scaling for
unknown parameter θ, one
single point
what is yoga
let's get closer to the values. However, such a pidkhid
can lead to rude pardons, and point
the assessment can be significantly revised from the true
the value of the estimated parameter (especially in
at the time of the vibirki small obyagu).

75. Butt

Solution. For my tasks, follow the assessment
two unknown parameters: a and σ2.
logarithmic likelihood function
may look

76.

Vіdkinuvshi in tsіy formulaі dodanok, which is not
deposit vіd a і σ2, store the system equal
credibility
Virishyuchi, otrimuemo:

77. CHAPTER 4

78.

(*)

79.

(*)

80.4.1. Evaluation of mathematical scaling of a normally distributed value for a given variance

vibirkove average
as the meaning of vipadkovo

81.

Maemo:
(1)
(2)

82.

(2)
(1)
(*)
(*)

83.4.2. Evaluation of mathematical scaling of a normally distributed value for unknown variance

84.

steps of freedom. Gustina

the value of є

85.

86. Student's rozpodіlu with n - 1 degrees of freedom

87.

88.

89.

know the formulas

90. 4.3. Estimation of the root-mean-square deviation of a normally distributed value

σ.

unfamiliar mathematical
points.

91. 4.3.1. Okremiya vіpadok vydomoy mathematicheskogo ochіkuvannya

Vicorist values
,


vibrational dispersion D*:

92.

magnitude
feel normal

93.

wash away
de
– width of rozpodіlu χ2

94.

95.

96.

97.4.3.2. Okremiya vpadok unknown mathematical ochіkuvannya

(devipad value


χ2 with n-1 steps of freedom.

98.

99.4.4. Evaluation of the mathematical scaling of the drop value for a fair selection

Choice of the great obsyagu (n >> 1).

100.

quantities
, what to wash

dispersion
, but otrimane
vibirkove average
yak value
vipadkovy size

magnitude
may be asymptotically


.

101.

wink the formula

102.

103.

Lecture 7

104.

Repetition of the past

105. CHAPTER 4

106.

Setting the estimation of the parameter of the input
rozpodіlu can virishuvati way
wake up the interval, in which of the given
Imovirnistyu use the reference value
parameter. This method of estimating
called interval assessment.
Call math for assessment
parameter θ will be uneven
(*)
de number δ characterizes the accuracy of the assessment:
which is less than δ, then the estimate is shorter.

107.

(*)

108.4.1. Evaluation of mathematical scaling of a normally distributed value for a given variance

Let's doslіdzhuvana vipadkovy value ξ rozpodіlena according to the normal law іz vіdomim
rms deviations σ and
nevіdomim math ochіkuvannyam a.
Required for the values ​​of the vibratory mean
evaluate mathematically scaling ξ.
Like before, looked at me obsessed
vibirkove average
as the meaning of vipadkovo
values, and the value of the selection x1, x2, …,
xn
rozpodіlenih nezalezhnyh vypadkovymi values
skin z yakikh maє mat. scaling a and mean square correction σ.

109.

Maemo:
(1)
(2)

110.

(2)
(1)
(*)
(*)

111.4.2. Evaluation of mathematical scaling of a normally distributed value for unknown variance

112.

Apparently, the value of tn is vipadical,
given such a rank, maє
Student's math k = n - 1
steps of freedom. Gustina
rozpodіlu imovіrnosti such
the value of є

113.

114. Student's rozpodіlu with n - 1 degrees of freedom

115.

116.

117.

Note. With a great number of steps
freedom k rozpodіl Student
pragne normal rozpodіlu s
zero mathematical scores and
single dispersion. Therefore, for k ≥ 30
dovirchy interval can be practical
know the formulas

118. 4.3. Estimation of the root-mean-square deviation of a normally distributed value

Let me finish the vipadka value
ξ subdivided according to the normal law
with mathematical points a that
unknown mean square
σ.
Let's take a look at two views: s vіdomim і
unfamiliar mathematical
points.

119. 4.3.1. Okremiya vіpadok vydomoy mathematicheskogo ochіkuvannya

Let me know the value M[ξ] = a i required
estimate only σ or the variance D[ξ] = σ2.
Let's guess what for the familiar mat. points
unbiased estimate of dispersion e
vibrational dispersion D* = (σ*)2
Vicorist values
,
appointed higher, introduced vipadkov
the value of Y, which accepts the value
vibrational dispersion D*:

120.

Let's take a look at the vipadian value
Sumi vipadkovі, what to stand under the sign
magnitude
feel normal
subdivided fN (x, 0, 1).
Todі Hn maє rozpodіl χ2 h n
steps of freedom like the sum of squares n
independent standard (a = 0, σ = 1)
normal vipadkovyh values.

121.

Significantly confidence interval h
wash away
de
– width of rozpodіlu χ2
and γ - superficiality (dovircha
imovirnist). The value of γ is numerically more
the area of ​​the shaded figure in fig.

122.

123.

124.

125. 4.3.2. Okremiya vpadok unknown mathematical ochіkuvannya

In practice, the situation is most aggravated,
if there are no offenses to the parameters of the normal
rozpodіlu: mathematical chіkuvannya a ta
root-mean-square variation σ.
I have a feeling of trust
the interval is based on the Fisher theorem,
Kit. next, what a vipad value
(devipad value
accepting meaning undisturbed
vibratory dispersion s2, may rozpodil
χ2 with n-1 steps of freedom.

126.

127.4.4. Evaluation of the mathematical scaling of the drop value for a fair selection

Interval evaluation of mathematical
grading M[ξ], subtracting for normal
razpodіlenoї vipadkoї value ξ ,
є, vzagali seeming, unsuitable for
vipadkovyh values
rozpodіlu. However, there is a situation, if
for any vipadkovy values ​​it is possible
koristuvatisya similar intervals
spіvvіdnoshnymi, - tse maє mіsce at
Choice of the great obsyagu (n >> 1).

128.

Like and more, let's look at the options
x1, x2,..., xn as independent values,
however, in different varieties
quantities
, what to wash
mathematical refinement M[ξi] = mξ i
dispersion
, but otrimane
vibirkove average
yak value
vipadkovy size
Appropriate to the central boundary theorem
magnitude
may be asymptotically
the normal law is subdivided c
mathematical scaling mξ and variance
.

129.

Therefore, we can see the value of dispersion
vipadkovy value ξ, then we can
flirting with approximate formulas
What is the value of the dispersion of the quantity ξ
it is not known that with great n it is possible
wink the formula
de s - corrected por_vn.-sq. mindfulness

130.

Repeated

131. CHAPTER 5. REVERSAL OF STATISTICAL HYPOTHESES

132.

A statistical hypothesis is a hypothesis about
looking at the unknown rozpodil or about the parameters
Vіdomogo rozpodіlu vipadkovoї size.
The hypothesis is reverified, what is meant to sound like
H0 is called the null or main hypothesis.
Dodatkovo vikoristovuetsya hypothesis H1,
to supersede hypotheses H0, called
competing with the alternative.
Statistical verification of hanging zero
hypotheses H0
vibratory tributes. For such a revision
pardons of two types are possible:
a) pardons of the first kind - vipadki, if you see
the hypothesis H0 is correct;
b) pardons of another kind - vipadki, if
the hypothesis H0 is accepted incorrectly.

133.

Imovirnist pardon of the first kind will be
name equal importance and designate
yak α.
The main method of reverification of statistical
hypotheses lie in the fact that
the actual selection is calculated values
statistical criterion - effective
the variable value of T, which is possible
I have broken the law. Area value T,
for which the main hypothesis H0 may
be judged, called critical, and
area value T, for which qu hypothesis
can be accepted, - the area is accepted
hypotheses.

134.

135.5.1. Rechecking the hypotheses about the parameters of the given parameters

5.1.1. Rechecking hypotheses about mathematics
normal
magnitude
Let the vipadic value ξ maє
normal rose.
It is necessary to reconsider the admission about those
what її mathematically clear one
real number a0. Let's look at the cream
fluctuations, if the dispersion of the house and if
out of nowhere.

136.

For different variance D[ξ] = σ2,
yak і in clause 4.1, significantly Vipadkov
the value that takes the value
vibratory mean. Hypothesis H0
back to back is formulated as M[ξ] =
a0. Oskіlki Vibrkove Middle
є unbiased estimate M[ξ], then
hypothesis H0 can be represented as

137.

Protection
vibratory variances, the null hypothesis is possible
write like this:
devipad value
accept the corrected value
the dispersion of the value of ξ i is similar to the variance
the value of Z discussed in Section 4.2.
How is the statistical criterion for vibero
vipadic value
I accept the meaning of the greater
vibrational dispersion to a minimum.

145.

Vipadkov value F maє
rozpodіl Fischer – Snedekor z
number of steps of freedom k1 = n1 – 1 and k2
= n2 – 1
more
corrected variance
, and n2 -
obsyag another choice, for which
smaller variance is known.
Let's look at two vidi competing
hypothesis

146.

147.

148. 5.1.3. Equalization of mathematical estimation of independent fluctuations

On the back of the head, we can look at the normal
rozpodіlu vipadkovyh values ​​v_domimi
dispersions, and then, on the basis of yoga,
drop of a fairly large difference in values ​​at
to finish the great independent vibirki.
Let the variable values ​​ξ1 and ξ2 be independent
divisions are normal, but let them have their variances D[ξ1]
and D[ξ2] vіdomi. (For example, stinks can be found
with any other knowledge of the insurance
theoretically). Vibrations n1 and n2
obviously. Come on
– vibirkovі
average for these vibes. Needed for vibrkovim
average at a given equal significance α
reverse the hypothesis about the validity of mathematical
assessment of the analysis of vipadical values ​​of growth from a priori mirkuvan,
Based on the minds of experiment, that
some excuse about parameters
rozpodіl dosl_dzhuyutsya, as shown
earlier. However, one often blames
the need to revise the mind
the hypothesis about the law was substantiated.
Statistical criteria, recognition
for such perevіrok, zazvychayutsya
meet the criteria.

154.

Vіdomo kіlka kіlka іїv sgodi. Perevagoyu
Pearson's criterion is yoga universality. 3 yoga
help you can change hypotheses about differences
law the rozpodіlu.
Pearson's criterion for bases on equal frequencies,
known by vibration (empirical frequencies), s
frequencies
to the law of rozpodіlu (theoretical frequencies)
Sound empirical and theoretical frequencies
fight. Slid z'yasuvati, chi vipadkovo
divergence of frequencies otherwise significant and explained
note that the theoretical frequencies of the calculation of the frequency of
incorrect hypotheses about rozpodіl general
marriage.
Pearson's criterion
nutrition, chi є zgoda hanging hypothesis
empirical data for a given equal
significance.

155. 5.2.1. Revisiting the hypothesis about normal rozpodil

Let's go є vipadkovy value ξ і zroblena
vibirka to do the great obliga n with more
kіlkіstyu different values variety. Needed
with equal significance of α, reverse the null hypothesis
H0 about those that the vipad value is divided
fine.
For clarity, the selection of the selection will take two numbers
α and β:
and divide the interval [α, β] into s
pіd_intervalіv. Let's take into account what is the meaning of the option,
what they drank in the leather interval, approximately equal
number that sets the middle of the interval.
Adjusted the number of options, which was taken to the skin Quantile of the order of α (0< α < 1) непрерывной
vipadkovy value ξ is called such a number xα,
for whom jealousy is won
.
The quantile x½ is called the median
quantities ξ, quantiles x0 and x2 - її quartiles, a
x0.1, x0.2,..., x0.9 - deciles.
For standard normal rozpodіlu(a=
0, σ = 1) i, also,
de FN (x, a, σ)
divergence variable value, and Φ(x) –
Laplace function.
Quantile of the standard normal distribution
xα for a given α can be known from the correlation

162.6.2. Rosepodil Student

Yakscho
– independent
vipadkovі values, scho mayut
normal rozpodіl іz nullovim
mathematical refinement and
single variance, then
rozpodіl vipadkovoї size
call Student's rose
with n steps of freedom (W.S. Gosset).

law of great numbers in the theory of similitudes, it is staggering that the empirical average (arithmetic mean) to finish the great final selection from the fixed distribution is close to the theoretical average (mathematical equivalence) of the distribution. The weak law of great numbers is stalely in view of zbіzhnosti, if there is room for zbіzhnist, imovirnosti, and the law of great numbers is strong, if there is room for zbіzhnіst, it may be everywhere.

Always know such a final number of trials, with some kind of predetermined imovirnistyu less 1 it is notable the frequency of the appearance of a deaco ї podії yak zavgodno little vіrіznyatiyetsya vіd yogo imovіrnostі.

Zagalny zm_st to the law of great numbers: spіlna diya great number to bring up the same and independent volatility factors to the result that it is impossible to fall into a slump.

To whom the power of the foundations of the method of assessing the fluency and improvement of the analysis of the final selection. A prime example is the prediction of the results of the selections based on the selection of the selections of the selections.

Encyclopedic YouTube

1 / 5

✪ Law of great numbers

✪ 07 – Theory of Imovirnosti. law of great numbers

✪ 42 Law of Great Numbers

✪ 1 - Chebishev's law of great numbers

✪ Grade 11, lesson 25, Gaussian curve. law of great numbers

Subtitles

Let's take a look at the law of great numbers, which, perhaps, is the most intuitive law in mathematics and the theory of immorality. And the shards of the wines stagnate to rich speeches; Let's start with the accuracy of I will give you a purpose, and then we'll talk about intuition. Let's take a vipadkovu value, for example X. Let's assume, we know її mathematically ochіkuvannya chi averaging for marriage. The law of great numbers just seems to be that we take the example of the n-th number of guards of a vipadkovy magnitude and we show the average number of all of them guards ... Let's take a change. We call її Х із the lower index n і з rice upland. The arithmetic mean of the nth number is the guardian of our vipadkovy value. My axis is more guarded. I will conduct the experiment once, and I will conduct the experiment once more, then I will conduct it again, and I will conduct the axis of caution, I will conduct it again, and I will omit the axis of the chain. I will conduct this experiment n-th number of times, and then I will add to the number of my warnings. The axis is my average value. The axis of the mean value of all guards, yak zrobila. The law of great numbers will tell us that my average vibratory will approach the mathematical scaling of the magnitude. Otherwise, I can also write that my vibratory average will be close to the average for the suupnistyu for n-ї kіlkost, which is not the case for inconsistency. I don’t clearly distinguish between the concepts of “neighborhood” and “comfort”, but I guess, if you intuitively understand that I’m going to take a big selection here, then I’m taking away mathematically for the totality as a whole. I think that most of you are intuitively wise, so I’ll get a sufficient number of trials with a great selection of applications, I’m allowed to try and give me a clearing of meanings, take it to the point of respect, mathematically clearing it up, and so on. Ale, I think, is often unaware of why it seems so. First, below, I will begin to explain why this is so, let me point out a specific example. The law of great numbers tells us what... Let's assume that we have a vipadkovy value of X. There's a lot of eagles with 100 deposits of the right coin. Nasampered, we know mathematically ochіkuvannya tsієї vypadkovoї value. Tse kіlkіst pіdkidan coin chi vyprobuvan, multiplied by the chances of success be-any viprobuvannya. So, tse one 50. That is the law of great numbers to say that we will take the test, otherwise I will bring the test to the average value, I will take it. .. First, if I spend the test, I drop a coin 100 times, or I take a box with a hundred coins, I’m afraid, and then I’ll blow it, I’ll take the number of eagles in me, and I’ll take away, for example, the number 55. It will be X1. Then I will re-string the box and subtract the number 65. Let me again - and subtract 45. I will test the number of times, and then I will test the number of times. The law of great numbers tells us that the average (the average value of all my guards) will be up to 50 in that hour, like n pragnite of incompetence. Now I would like to talk a little bit about those who feel this way. It’s good to know that even after 100 trials, my result is higher than the average, then according to the laws of flexibility, I can have more or less eagles in order to, so be it, compensate for the difference. We do not call those who will become. Tse is often called the "pardon of gambling engraving." Let me show you the difference. I vikoristovuvatimu coming butt. Let's draw a graph. We remember color. Tse n, my weight X - Tse n. Tse kіlkіst viprobuvan, yakі I will spend. And my whole Y will be the middle one. We know that mathematically scaling is more than enough 50 years old. Let's paint. Tse 50. Let's turn to our butt. It’s not so bad… For the first hour of my first test, I took 55, but that’s my average. I have less than one data entry point. Then, after two tests, I will take away 65. Later, my average will be 65 + 55, subdivided by 2. Tse 60. And my average is three times over. Then I subtracted 45, which again lowered my arithmetic mean. I don't put 45 on the chart. Now I need to bring everything to the average value. Why is 45+65 worth it? Let me break the value, to mark the point. Ce 165 dility by 3. Ce 53. Hi, 55. Later, the average value again drops to 55. We can continue testing. Since we have made three trials and taken away the middle ones, there are many people to think that the gods of omnipotence should work in such a way that we have less eagles in the future, that in the coming trials the results will be lower, so that the average value will change. Ale tse zavzhd so. Nadal ymovіrnіst zavzhaєєєєєєє such itself. The mobility of what is an eagle in me will be 50%. Not those who in me fall a little more than a small number of eagles, more, lower I check, but far away raptly turn tails. Tse "pardon engraving." Just because you have an unbearably large number of eagles, it doesn’t mean that at the singing moment you have an overwhelmingly large number of tails. We don't call it that. The law of great numbers tells us that it cannot be significant. Let's say, after the last kіlkost's number of tests, your average ... The ability to finish it is small, prot... Let's say, your average has reached the number of marks - 70. You think: “Wow, we looked like a mathematician smartly.” Ale, the law of great numbers, it seems, to you, baiduzhe, we have tried several tests. We have all the same lost an inexhaustible amount of testing in front. Mathematically ochіkuvannya tsієї innumerable number of tests, especially in similar situations, will come. If you come before the last date, as if turning around great value, the infinite number that goes with it, I will recreate it to a mathematical refinement. Tse, zvichayno, even more gloomy, ale tse those that seem to us the law of great numbers. Tse is important. It doesn’t seem to us that we had a lot of eagles, then it’s possible that the resolution of the resolution will increase in order to compensate. Whose law is to tell us that it doesn’t matter, what a result with the last number of trials, because you still have lost an infinite number of trials in advance. And as soon as you grow enough of them, you will turn again to mathematical refinement. Tse important moment. Think new. But we don’t win today in practice with lotteries and in the casino, if we want to see that you still have a sufficient amount of sampling ... We can probably get away with it ... why do we care about the quality of what we seriously believe in the norm? But casinos and lotteries are now working on this principle, which is to take a sufficient number of people, naturally, in a short term, with a small sample, a few people to win the jackpot. Ale, for the great term, the casino will forever lose money from the winner through the parameters of the game, and ask you to get paid from the stink. This is the important principle of imovirnosti, which is intuitive. If you want to, if you formally explain it to you with vipadkovymi values, everything seems to be confused. Everything that the law says, the more vibes, the more arithmetic mean of these vibrok pragnime to the right mean. And if you are more specific, then the arithmetic mean of your selection will be matched with the mathematical estimates of the magnitude. From i all. See you next video!

The Weak Law of Great Numbers

The weak law of great numbers is also called Bernoulli's theorem, after Jacob Bernoulli, who proved it in 1713.

Let there be an inconsequential sequence (succession of recurrence), however, in divisions and uncorrelated fluctuations. That is the covariance c o v (X i , X j) = 0, ∀ i ≠ j (\displaystyle \mathrm (cov) (X_(i),X_(j))=0,\;\forall i\not =j). Come on. Significantly through the average of the first n (\displaystyle n) members:

.

Todi X n → P μ (\displaystyle (\bar (X))_(n)\to ^(\!\!\!\!\!\!\mathbb (P) )\mu ).

Tobto for be-something positive ε (\displaystyle \varepsilon)

lim n → ∞ Pr (| X n − μ |< ε) = 1. {\displaystyle \lim _{n\to \infty }\Pr \!\left(\,|{\bar {X}}_{n}-\mu |<\varepsilon \,\right)=1.}

Strengthening the law of great numbers

Come on, there is an inexhaustible sequence of independent, however, varying subdivisions ( X i ) i = 1 ∞ (\displaystyle \(X_(i)\)_(i=1)^(\infty )), appointed on one ymovirnіsny expanse (Ω , F , P) (\displaystyle (Omega ,(\mathcal (F)), \mathbb (P))). Come on E X i = μ , ∀ i ∈ N (\displaystyle \mathbb (E) X_(i)=\mu ,\;\forall i\in \mathbb (N) ). Significantly through X n (\displaystyle (\bar(X))_(n)) vibirkove middle of the first n (\displaystyle n) members:

X n = 1 n ∑ i = 1 n X i , n ∈ N (\displaystyle (\bar(X))_(n)=(\frac(1)(n))\sum \limits _(i= 1 )^(n)X_(i),\;n\in \mathbb (N) ).

Todi X n → μ (\displaystyle (\bar (X))_(n)\to \mu ) please wait.

Pr (lim n → ∞ X n = μ) = 1. right) = 1.) .

As if it were a mathematical law, the law of great numbers, it can become real light only with the help of allowances, it can only be achieved with the singing world of accuracy. So, for example, wash the last samples often cannot be saved indefinitely for a long time and with absolute accuracy. In addition, the law of great numbers is less to be said about naming significant value of the average value of the mathematical value.

The function of rozpodіlu vipadkovoї magnitude and її power.

The function was subdivided The function F(X) of the vipadical value of X is called the function F(X), which shows for skin x the ability of the vipadical value of X to take a value less than x: F(x)=P(X

Function F(x) name different integral function rozpodіlu abo іntegral law rozpodіlu

The power of the function was subdivided:

1. The function of subdividing a variable value is an invisible function, placed between zero and unity:

0 ≤ F(x) ≤ 1.

2. The function of divergence of a variable value is an invariable function along the entire numerical axis.

3. At the minus of inconsistency, the function rose to zero, plus infinity to one, so: F(-∞)= , F(+∞)= .

4. Probability of dropping the value up to the interval [x1,x2) (including x1) to increase the growth of the function according to this interval, tobto. P(x 1 ≤ X< х 2) = F(x 2) - F(x 1).

Uncertainty of Markov and Chebisheva

Markov's unevenness

Theorem: Even though the value of X takes on only inconspicuous values ​​and can be mathematically refined, then for any positive number A the equality is correct: P(x>A) ≤ .

Since X > A and X ≤ A are protracted, then replacing P (X > A) it is possible to turn 1 - P (X ≤ A), we arrive at a different form of Markov's unevenness: P (X ≥ A) ≥1 - .

The unevenness of Markov zastosovuetsya to be-any nevid'emnyh vipadkovyh values.

Chebishev's nervousness

Theorem: For whether there is a reversed value, such as a mathematical calculation of that variance, Chebishev's inevitability is valid:

Р (|Х – a| > ε) ≤ D(X)/ε 2 or Р (|Х – a| ≤ ε) ≥ 1 – DX/ε 2 de a = M(Х), ε>0.

The law of great numbers "at the form" of Chebishev's theorem.

Chebishev's theorem: How much dispersion n independent vipadic values ​​Х1, Х2,…. X n fringed one and the same and constant, then with an unfenced increase in the number n the arithmetic mean of the falling values ​​converge after the arithmetic mean of the mathematical refinement a 1 ,a 2 ….,a n, then .

The sense of the law of great numbers is based on the fact that the average values ​​of the falling values ​​can be calculated from the mathematical calculation at n→ ∞ imovirno. Vіdhilennya avіdnіh vіd mathematical ochіkuvannya staє skіlki zavgodno small z ymovirnіstyu, close to one, yakscho n dosit big. In other words, the possibility of any change in the average values a the skils are always small for the sizes n.

30. Bernoulli's theorem.

Bernoulli's theorem: Part of the way to n repeated independent trials, in dermal diseases, it may be possible for one and the same time to be removed, with an unrestricted increase in the number n converge for imovirnistyu to imovirnosti r tsієї podії in okremu testing: \

Bernoulli's theorem is a legacy of Chebishev's theorem, because part of the subdivision is possible as the arithmetic mean of n independent alternative variable values, the same law can be subdivided.

18. Mathematical evaluation of discrete and non-stop fluctuations and their power.

Mathematical points the sum of creations of all is called

For a discrete drop-down quantity:

For an uninterrupted drop-down value:

The power of mathematical refinement:

1. Mathematical evaluation of the constant value of the most recent value: M(S)=S

2. The constant multiplier can be blamed for the sign of mathematical refinement, so M(kX)=kM(X).

3. Mathematical grading of the algebraic sum of the final number of variable values ​​is just the sum of їх mathematical grading, that is. M(X±Y)=M(X)±M(Y).

4. Mathematical grading to the creation of the last number of independent variable values ​​is the same as the creation of their mathematical grading: M(XY)=M(X)*M(Y).

5. If all the values ​​of the vipadkovoї value are changed (changed) by a constant, then by the qiu of the constant Z zbіlshitis (changes) mathematically ochіkuvannya tsієї vypadkovoї value: M(X±C)=M(X)±C.

6. Mathematical grading of the variable value in the form of її mathematical grading to zero: M=0.