Independent sizes. Operations over vypadkovym values. Fallow lands and squares of sizes Independent sizes of butts

& nbsp Fallow lands and squares of size

& NbspIt is necessary to respect the steps and nature of their fallowness for the development of systems of large values. Qia depletion can be more and more tough, more tough. In some cases, the fallowness of the same magnitude can be flat, but, knowing the value of the same magnitude, it can be specified in accuracy. In the extreme drop, the fallowness of the area is very low and very small, but it can practically be covered by the neighboring ones.
& NbspUnderstanding about independent values - one of the most important to understand the theory of imperialities.
& Nbsp A random variable \ (Y \) is called an independent type of a type of value \ (X \), since the law of a rise in the value \ (Y \) does not lie as the value of \ (X \) has taken on.
& Nbsp \).
& NbspNot_v, if \ (Y \) is in \ (X \), then $$ f (y \ mid x) \ neq f_ (2) (y) $$ & nbsp fallowness or indeterminacy of large values depends on each other: If the value \ (Y \) does not lie in \ (X \), then the value \ (X \) does not lie in \ (Y \).
& NbspReally, don’t let \ (Y \) not lie in \ (X \): $$ f (y \ mid x) = f_ (2) (y) $$ maєmo: $$ f_ (1) (x) f ( y \ mid x) = f_ (2) (y) f (x \ mid y) ...
& NbspSo, both the fallowness and the indefiniteness of the large values are dependent on each other, it is possible to date the new values of the independent large values.
& NbspThe random quantities \ (X \) і \ (Y \) are called squares, because the law of dermal growth from them does not lie in the fact that the meaning was accepted by the insha. In general, the values \ (X \) і \ (Y \) are called fallow.
& Nbsp For independent continuous values, the theorem of multiple laws in the growth of a vigle: $$ f (x, y) = f_ (1) (x) f_ (2) (y) $$ i.e. the range of values to be included in the system.
Often, according to the most respectable function \ (f (x, y) \), it is possible to create patterns, but the values \ (X, Y \) є are independent, but the same, as the distribution \ (f (x, y) \) falls into there are two functions, for which one lie only from \ (x \), іnsha - only from \ (y \), then the size of the square.
Butt 1. The distribution of the system \ ((X, Y) \) ma view: $$ f (x, y) = \ frac (1) (\ pi ^ (2) (x ^ (2) + y ^ (2) + x ^ (2) y ^ (2) +1)) $$
Decision. You can fold the denominator into factors, mєmo: $$ f (x, y) = \ frac (1) (\ pi (x ^ (2) +1)) \ frac (1) (\ pi (y ^ (2) +1 )) $$ In addition, the function \ (f (x, y) \) has been split into two functions, one of which is only from \ (x \), and іnsha - only from \ (y \), robimo visnovok, whose values \ (X \) and \ (Y \) are guilty of being independent. Simple, stashed formulas, mєmo: $$ f (x, y) = \ frac (1) (\ pi (x ^ (2) +1)) \ int _ (- \ infty) ^ (\ infty) (\ frac (dy) (\ pi (y ^ (2) +1))) = \ frac (1) (\ pi (x ^ (2) +1)) $$ similar to $$ f (x, y) = (\ frac (1) (\ pi (y ^ (2) +1))) $$ crossover stars, u $$ f (x, y) = f_ (1) (x) f_ (2) (y) $$ i , also, the values \ (X \) і \ (Y \) are independent.

Two types of values of $ X $ and $ Y $ are called independent, as the law of growth of one type of value does not change because of the fact that a possible value has taken the value of a certain type. So, for any $ x $ і $ y $ the subії $ X = x $ і $ Y = y $ є are not adjacent. Oskilki podії $ X = x $ і $ Y = y $ independent, then according to the theorem, create the dimensions of the independent pods $ P \ left (\ left (X = x \ right) \ left (Y = y \ right) \ right) = P \ left (X = x \ right) P \ left (Y = y \ right) $.

butt 1 ... Come on, the $ X $ value is a penny win for the tickets of the one "Russian Lotto" lottery, and the $ Y $ value is a penny win for the tickets from the Golden Key lottery. Obviously, the values of $ X, \ Y $ will not be equal, so as the game for the tickets of the same lottery does not lie in the law of the distribution of votes for the tickets of the first lottery. In addition, if the values of $ X, \ Y $ were picked up by the same lottery, then, obviously, the data on the values of the boules would be fallen.

butt 2 ... Two robots work in the small workshops and prepare the development of virobi, not connected with other technologies for the preparation and vicorization of the syrovin. The law of the rise in the number of defective virobes, who were prepared by the first robots for a change, such a viglyad:

$ \ Begin (array) (| c | c |)
\ hline
Number \ defective \ virob_v \ x & 0 & 1 \\
\ hline
Imoviness & 0.8 & 0.2 \\
\ hline
\ End (array) $

The number of defective virobes, prepared by other workers for a change, is ordered by the offensive to the law of growth.

$ \ Begin (array) (| c | c |)
\ hline
Number \ defective \ virobiv \ y & 0 & 1 \\
\ hline
Imoviness & 0.7 & 0.3 \\
\ hline
\ End (array) $

We know the law of the rise in the number of defective virobes, prepared by two robots for a change.

Don't worry, the value of $ X $ is the number of defective virobes that were prepared by the first worker for a change, and $ Y $ is the number of defective viruses that were prepared for another robot for a change. Behind the sinking, in the amount of $ X, \ Y $ are independent.

The number of defective virobes prepared by two workers for a change is $ X + Y $. The possible values are $ 0, \ 1 $ and $ 2 $. It is known that the value of $ X + Y $ takes its value.

$ P \ left (X + Y = 0 \ right) = P \ left (X = 0, \ Y = 0 \ right) = P \ left (X = 0 \ right) P \ left (Y = 0 \ right) = 0.8 \ cdot 0.7 = 0.56. $

$ P \ left (X + Y = 1 \ right) = P \ left (X = 0, \ Y = 1 \ or \ X = 1, \ Y = 0 \ right) = P \ left (X = 0 \ right ) P \ left (Y = 1 \ right) + P \ left (X = 1 \ right) P \ left (Y = 0 \ right) = 0.8 \ cdot 0.3 + 0.2 \ cdot 0.7 = 0.38. $

$ P \ left (X + Y = 2 \ right) = P \ left (X = 1, \ Y = 1 \ right) = P \ left (X = 1 \ right) P \ left (Y = 1 \ right) = 0.2 \ cdot 0.3 = 0.06. $

That law raises the number of defective virobes prepared by two robots for a change:

$ \ Begin (array) (| c | c |)
\ hline
Number of \ defectives \ viruses & 0 & 1 & 2 \\
\ hline
Imovirnost & 0.56 & 0.38 & 0.06 \\
\ hline
\ End (array) $

In front of the front butt, an operation was performed on the large quantities $ X, \ Y $, and the sum $ X + Y $ itself was known. Damo now more suvore value of operations (additional, growth, multiplicity) over large values and put the solution, as it should be.

value 1... The creator of $ kX $ of the value $ X $ for the post-value of $ k $ is called the value of $ kx_i $, with the same options $ p_i $ $ \ left (i = 1, \ 2, \ \ dots, \ n \ right) $.

value 2... The sum (difference) of the large quantities $ X $ і $ Y $ is called the vipadkov quantity, which can be taken in the form $ x_i + y_j $ ($ x_i-y_i $ or $ x_i \ cdot y_i $), de $ i = 1 , \ 2, \ dots, \ n $, if $ p_ (ij) $ is the same if $ X $ is equal to $ x_i $, and $ Y $ is $ y_j $:

$$ p_ (ij) = P \ left [\ left (X = x_i \ right) \ left (Y = y_j \ right) \ right]. $$

Since the values $ X, \ Y $ are independent, then, according to the theorem, the multiplicity of values for the independent pods is: $ p_ (ij) = P \ left (X = x_i \ right) \ cdot P \ left (Y = y_j \ right) = p_i \ cdot p_j $.

butt 3 ... Independent types of values $ X, \ Y $ are given by their own laws of distribution of values.

$ \ Begin (array) (| c | c |)
\ hline
x_i & -8 & 2 & 3 \\
\ hline
p_i & 0.4 & 0.1 & 0.5 \\
\ hline
\ End (array) $

$ \ Begin (array) (| c | c |)
\ hline
y_i & 2 & 8 \\
\ hline
p_i & 0.3 & 0.7 \\
\ hline
\ End (array) $

Similarly, the law of the distribution of the value $ Z = 2X + Y $. The sum of the small values $ X $ і $ Y $, that is, $ X + Y $, is called a specific value, which can be taken in all possible values in the form $ x_i + y_j $, de $ i = 1, \ 2, \ dots, \ n $, from the top of $ p_ (ij) $, if the value $ X $ is the same as the value of $ x_i $, and $ Y $ is the value of $ y_j $: $ p_ (ij) = P \ left [\ left (X = x_i \ right) \ left (Y = y_j \ right) \ right] $. Since the values $ X, \ Y $ are independent, then, according to the theorem, the multiplicity of values for the independent pods is: $ p_ (ij) = P \ left (X = x_i \ right) \ cdot P \ left (Y = y_j \ right) = p_i \ cdot p_j $.

Also, the laws of the distribution of the $ 2X $ and $ Y $ values are different.

$ \ Begin (array) (| c | c |)
\ hline
x_i & -16 & 4 & 6 \\
\ hline
p_i & 0.4 & 0.1 & 0.5 \\
\ hline
\ End (array) $

$ \ Begin (array) (| c | c |)
\ hline
y_i & 2 & 8 \\
\ hline
p_i & 0.3 & 0.7 \\
\ hline
\ End (array) $

For the sake of convenience, all the values of the sum $ Z = 2X + Y $ and the number of values of the sum of multiplication of the values of the given values of the same values $ 2X $ і $ Y $.

As a result, we can take out $ Z = 2X + Y $:

$ \ Begin (array) (| c | c |)
\ hline
z_i & -14 & -8 & 6 & 12 & 10 & 16 \\
\ hline
p_i & 0.12 & 0.28 & 0.03 & 0.07 & 0.15 & 0.35 \\
\ hline
\ End (array) $

Vypadkovі podії are called independent, as when one of them appears, it doesn’t add to the quality of appearing podіy.

butt 1 . If there are two or more urns with color kuli, then the knight of what kind of kuli from one urn does not fit into the ability to reject the other kuli from the resolves of urns.

For independent pod_y is fair multiplicity theorem:(one hour)show the decilkokh independent vipadkovyh podіvnyu additional additonalities:

Р (А 1 і А 2 і А 3 ... і А k) = Р (А 1) ∙ Р (А 2) ∙ ... ∙ Р (А k). (7)

Sleeping (one hour) when you come up with a sign means that you can see it A 1,і A 2,і A 3... і A k.

butt 2 . Є two urns. In one there are 2 black and 8 big bags, in the first there are 6 black and 4 big ones. hello go A-vibir navmannya biloi kuli from the first urn, V- from the other. Yaka ymovіrnіst vibraty navmannya simultaneouslyі from cikh urns on a bіlіy kulі, so as to what to drive R (Aі V)?

Decision: ymovіrnіst dіstati bіla kul from the first urn
R(A) = = 0.8 s other - R(V) = = 0.4. Imovirnist one hour by the way on a big cooler from both ballot boxes -
R(Aі V) = R(A)· R(V) = 0,8∙ 0,4 = 0,32 = 32%.

Application 3. Ratio of reduction in iodine intake in 60% of the animals of the great population. For the experiment, 4 extra zolos are required. Know the value of the fact that 4 vypadkovo-faced creatures will have a shield-like zloza.

Decision: Vipadkova podіya A- Vibir navmannya creature with zbіlsheny thyroid zloza. For the mindset of the podiatry R(A) = 0.6 = 60%. The type of spiral appearance of the choice of independent pod_y - vibration of 4 creatures with a healthy shield-like hair - will be suitable:

R(A 1 i A 2 i A 3 i A 4) = 0,6 ∙ 0,6 ∙0,6 ∙ 0,6=(0,6) 4 ≈ 0,13 = 13%.

Calling podії. The multiplication theorem for long-lying pods

Vypadkovy subtypes A and B are called fallow lands, as there are one of them, for example, And the change in the nature of the emerging subtypes - V. There are two meanings of ymoviness for the fallow pods: unconditional and conditional probabilities .

yaksho Aі V depending on the situation, then on the basis of the current situation V persim (tobto to go A) be called crazy ymovirnistyu tsієї podії and know R(V) Ymovіrno nastannya podії V for wash, come on A already seen, be called mindfulness podії V i know R(V/A) abo P A(V).

The analogous sense may be insane - R(A) І cleared - R(A / B) Ymovіrnostі for podії A.

The theorem of multiplication of probabilities for two fallow pods: the efficiency of one-hour setting of two fallow pods And in addition to the madness of the first pod on the mind of another:

R(And that B)= P(A)∙ R(B / A) , (8)

A, abo

R(And that B)= P(V)∙ R(A / B), (9)

how to persimmon infusion V.

Butt 1. In the urn there are 3 black cakes and 7 big ones. Know the value of the fact that there will be 2 big balls from the chain of urns (and the first ball cannot be turned into the urn).

Decision: Ymovіrnіst dіstatі the first bіla kulya (podіya A) Dorіvnyuє 7/10. For that yak vіn viymut, 9 sacks will be left in the urn, of which 6 bіlich. Todi ymovirnist appear another bіlogo kulі (podіya V) door R(V/A) = 6/9, and by the way, it was two big ones.

R(Aі V) = R(A)∙R(V/A) = = 0,47 = 47%.

The theorem of multiplication of parameters for fallow pods has been introduced. Zokrema, for three pod_y, tied one by one:

R(Aі Vі Z)= P(A)∙ R(B / A)∙ R(Z / AV). (10)

Butt 2. In two children's gardens, skinned from each side of 100 children, sleeping in infectious diseases. Parts of ailments become approximately 1/5 and 1/4, moreover, in the first mortgage 70%, and in the other - 60% of the ailments - children up to 3 rocks. Vypadkovym rank select one child. First of all, you need to know how:

1) the child's turn to be carried to the first child's cage (pod_ya A) I ailment (pod_ya V).

2) turn the child from another baby cage(pod_ya Z), Khvoriy (pod_ya D) І older than 3 rock_v (pod_ya E).

Decision. 1) shukana ymovirnist -

R(Aі V) = R(A) ∙ R(V/A) = = 0,1 = 10%.

2) shukana ymovіrnіst:

R(Zі Dі E) = R(Z) ∙ R(D/C) ∙ R(E/CD) = = 5%.

Bayes' formula

= (12)

Butt 1. When first looking at a sick person, 3 diagnoses are transmitted H 1 , H 2 , H 3.Oh ymovirnosti, on the thought of the doctor, are distributed as follows: R(H 1) = 0,5; R(H 2) = 0,17; R(H 3) = 0.33. From the same time, the first diagnosis is the most important. For this clarification, it is meant, for example, an analysis of blood, in which one can see the improvement of the SHOE (podiya A). Late in view (the results are up to the last), but the change in SHOE in case of transfer of ill-fated children:

R(A/H 1) = 0,1; R(A/H 2) = 0,2; R(A/H 3) = 0,9.

In the discarded analysis, the revised SHOE (pod_ya A added). Todi rosrahunok for Bayes' formula (12) give the value of the ymovirnosti perebachuvannyh ailments with an increase in the value of SHOE: R(H 1 /A) = 0,13; R(H 2 /A) = 0,09;
R(H 3 /A) = 0.78. The figures show that from the laboratory data the most real is not the first, but the third diagnosis, the quality of what has now appeared to be a great one.

Appendix 2. Significance and assessment of the risk of perinatal * mortality of the child in women with anatomically narrow pelvis.

Decision: Come on go H 1 - safe canopy. For the data of key events, R(H 1) = 0.975 = 97.5%, todi, where H 2- the fact of perinatal mortality, then R(H 2) = 1 – 0,975 = 0,025 = 2,5 %.

meaningfully A- the fact of the presence of the high pelvis in the breed. During the follow-up visits: a) R(A/H 1) - the size of the high pelvis with friendly canopy, R(A/H 1) = 0.029, b) R(A/H 2) - the size of the pelvis during perinatal mortality,
R(A/H 2) = 0.051. Todi shukan ymovіrnіnіrіst of perinatal mortality in vzіku pozі in bredіllі should be insured by the formula Bays (12) and dorіvnyu:

Thus, the risk of perinatal mortality in the anatomically high pelvis is significant (mayzhe vdvichi) of the middle rhizik (4.4% versus 2.5%).

When vivchenny systems of vypadkovyh values depend on the animals, I respect the steps and the nature of their fallowness. The amount of fallowness can be more and more twisted, more and more thick. In some cases, the fallowness of the same magnitude can be flat, but, knowing the value of the same magnitude, it is possible to specify the value of the same. In the extreme drop, the fallowness of the area is very low and very small, but it can practically be covered by the independent ones.

Understanding about independent values is one of the most important ones to understand the theory of definitions.

Vypadkova value is called an independent type of vypadkova value, as the law of a rise in value does not lie as a value has taken on a value.

For non-perennial values of the mind of independence, it can be written in the viewer:

at be-yaku.

Navpaki, in times, if you lie down, then

Evidently, the fallowness or the indefiniteness of the lower values is dependent on each other: as the value does not lie in.

Dіysno, don’t let it go:

. (8.5.1)

From formulas (8.4.4) and (8.4.5) maєmo:

stars, beruchi to uvaga (8.5.1), otrimaєmo:

it’s necessary to bring it.

Thus, both the fallowness and the indifference of the large values are dependent on each other, it is possible to date the new values of the independent large values.

Vypadkovі values і are called independent, as the law of the growth of the skin from them does not lie in the way that the meaning has taken the insha. In the least, the value is called fallow.

For non-overlapping irreversible values, the theorem of multiplying laws in the growth of the growth is:

, (8.5.2)

that is, the scope of the distribution of the system of independent variable values is also additional to the distribution of the distribution of the same values, which should be included in the system.

Umov (8.5.

Often, according to the most respectable function, it is possible to create a variety of different sizes, but not too large ones, but, as a matter of fact, there are two functions that fall into two functions, from which one can only be found, only small ones.

Butt. The power of the system mak viglyad:

Visually, deserted or independent sizes of i.

Decision. Fold the banner into multipliers, maєmo:

Moreover, the function has been expanded into two functions, one is fallen only from one side, and one is from one side only from one side, and one is from one another, robimly from one to another, which is of the magnitude and is to blame for being independent. Dіysnо, zasosovuchi formulas (8.4.2) і (8.4.3), maєmo:

;

similarly

perekonumosya, scho

і, also, the size і square.

Visceral possessions criterion of judgment about fallowness, or indeterminacy of low values, should be taken into account, which is the law of development of systems and we see them. On the practical side of the buvak navpaki: the law of the development of systems and not vidoms; vіdomі only the laws of the distribution of the limits of the values that enter into the system, and for the sake of the value and the independent. It is possible to write the status of the distribution of the system as the setting of the distribution of the values to enter into the system.

We make a cheap lecture on an important understanding about the “fallowness” and “independence” of large quantities.

The understanding of the "independency" of large quantities, which is the case in the theory of the ideas of values, can be easily understood from the common understanding of the "deposition" of quantities, which are operatively in mathematics. Dіysnо, tune for "fallowness" values may only be one type of fallowness - povnu, zhorstku, so called - functional fallowness. Two quantities are called functionally fallow, which, knowing the meaning of one of them, can precisely be called the meaning of one.

In the theory of imovity, the formation of a large, large, type of fallowness - of an imovirny or "stochastic" fallowness. If the value is tied to the size of the quantity, then, I know the value, it is not possible to specify the exact value, but it is possible to impose only the law of growth, so that it can be deposited because the value has taken on the value.

Imovirnіsna fallowness can be more-less dense; in the world of increasing density, the density of deposits is getting closer and closer to functional. In such a rank, the functional fallowness can be seen as an extreme, boundary type of the most probabilistic fallowness. The Інshi extreme vypadok - on the whole the indelibility of vypadkovyh values. Between the two, in extreme weather, all the gradations of the fallowing ones lie - from the strongest to the strongest. These physical quantities, which, in practice, are important functionally fallow, in reality, depending on the size of large quantities of fallow: with a given value of one of the number of quantities, it is important to grow in practice in practice. On the other side, those sizes, which are important in practice, are quite independent, and for some reason they often endure in many cases of fallowness, although the fallowness of the surface is weak, but it can be used for practical purposes.

Imovirnіsnіsnіsnіvіnіtіvіvіvіvіvnymi values even more often be studied in practice. It doesn’t mean that it’s big and small, but it doesn’t mean that it’s a change in magnitude; it does not mean that due to a change in value, the value of a tendency also changes (for example, growth or decline during growth). Qia tendency to see deprivation "in the middle", in wild rice, І in the skin envelope from no way out.

It is understandable, for example, two such vypadkovy sizes: - the growth of the taken people, - the yogh. Obviously, the magnitude and are found in the singularity of the fallow; vona turn around in that, in the zagalny, people with great zrostannyy mayut more vagu. It is possible to find the empirical formula, which will closely replace the functional abundance. Such, for example, is the formula for the home, it is close to the turn of the fallowness between the growth and the wagon.

PODI VIPADKOVI NEZALEZHNI - such vipadkovi podії Aі V, for yak value P one hour nastaniya 2 podіy A to B additional information about the quality of the skin conditions for them: P (AB) = P (A) P (B). Similarly to the value of independence NS vypadkovyh pod_y. The price will expand to the independence of low values, but itself, vypadkovі values X 1, X 2, ..., X n square, like for any group X i1, X i2, ..., X ik, the number of values is true parity: P (X i1 ≤ x i1, X i2 ≤ x i2, ..., X ik ≤ x ik) = P (X i1 ≤ x i2) P (X i2 ≤ x i2) ... (P (X ik ≤ x ik); 1≤ k ≤ n. When the geol. problems by the methods of the theory of imovings and mathematical statistics Correct fallowness of pre-matured values is often the best folding and generalized part of pre-production.

Geological vocabulary: in 2 volumes. - M .: Nedra. Edited by K. N. Paffengolts and I.. 1978 .

Marvel at the same "PODIS VIPADKOVI NEZALEZHNI" in the following dictionaries:

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