The result is more easily presented as Table 9.6.

Table 9.6

Accelerated by formulas for cob moments and results in tables 9.3 and 9.4 and numerically mathematical calculation of warehouse Xі Y:

Dispersions can be calculated through another corn moment and the results of the table. 9.3 and 9.4:

To calculate the covariance Before(X,Y) vikoristovuemo similar formula through the cob moment:

The correlation coefficient is determined by the following formula:

Shukana imovirnіst is defined as imovіrnіst hitting the area on the flat, which is signified as a type of unevenness:

9.2. The ship is transmitting SOS messages, which may be received by two radio stations. This signal can be received by one radio station independently from outside. Imovirnist that the signal of acceptance by the first radio station becomes 0.95; the ability of the signal to be received by another radio station is more than 0.85. To know the law of distribution of the two-world vipadkovy value, which characterizes the reception of a signal by two radio stations. Write a function to split.

Solution: Come on X- Podiya, yak I guess that the signal is received by the first radio station. Y- Podіya polagaє that the signal is received by another radio station.

Anonymous meaning .

X=1 - signal of acceptance by the first radio station;

X=0 – the signal was not received by the first radio station.

Anonymous meaning .

Y\u003d l - signal of acceptance by another radio station,

Y=0 - signal not accepted by another radio station.

The mobility of the fact that the signal is not accepted by either the first or the other radio stations is more expensive:

Imovirnіst priynyattya signal to the first radio station:

Imovirnist that the signal is received by another radio station:

Imovirnist of the fact that the signal is received by the first and the other radio stations, dorovnyuє: .

Then the law of distribution of the two-world vipadkovy value is more expensive:

X,y) value F(X,y) a good sum of abilities and possible values ​​of the magnitude X,Y);

The same function rozpodіlu mother looked:

9.3. Two companies release the same products. The skin is independent in another way, you can praise the decision about the modernization of virobnitstva. Imovirnist that the first company praised such a solution, good 0.6. The ability to accept such a decision by another company is worth 0.65. Write the law of distribution of the two-world vipadkovoy value, which characterizes the praise of the decision about the modernization of the production of two firms. Write a function to split.

Suggestion: The law was subdivided:

With skin fixed value, points with coordinates ( x,y) the meaning of the sum of the values ​​of the quiet possible values, as if they were trapping the middle of the indicated rectangle .

9.4. Piston rings for car engines are made at the automatic lathe. Tovshchina kіltsya (vipadkovy value X)і opening diameter (vipad value Y). Apparently, close to 5% of all piston rings are defective. Moreover, 3% of the boat is equipped with non-standard diameters of opening, 1% - with non-standard tovshchina and 1% - is rejected for both signs. Know: splny rozpodіl dvovimіrnoї vipadkovї size ( X,Y); odnomirnі rozpodіlu warehousing Xі Y;mathematical cleaning of warehouses Xі Y; correlation moment and coefficient of correlation between warehouses Xі Y two-dimensional vipadkovy value ( X,Y).

Suggestion: The law was subdivided:

; ; ; ; ; .

9.5. The products to the factory have a heritage of defects BUT become 4%, and after the defect At- 3.5%. Standard products stock 96%. Indicate if any product may have defects and both types.

9.6. Vipadkov value ( X,Y) razpodіlena z postіynoyu shіlnistyu middle of the square R, The vertices of which can be coordinated (-2; 0), (0; 2), (2; 0), (0; - 2). Calculate the thickness of the rose under the vertical size ( X,Y)і umovnі shіlnostі rozpodіlu R(X\at), R(at\X).

Solution. Let's stay on the square x 0y tasks square (Fig.9.5) and significantly equal sides of the square ABCD, accelerating straight lines that pass through two given points: Substituting the coordinates of the vertices BUTі At we take sequentially equal sides AB: or .

Similarly, we know the equivalence of the parties ND: ;side CD: i sides DA: . : .D X , Y) is a pіvkulyu with the center on the cob of coordinates to the radius R.Know the scope of the difference between emotions.

Suggestion:

9.10. A discrete two-world variable value is given:

Know: a) the mental law has been subdivided X, mind you, what y= 10;

b) the mental law has been subdivided Y, mind you, what x =10;

c) mathematical refinement, dispersion, correlation coefficient.

9.11. Uninterrupted two-world vipadic value ( X,Y) equally divided in the middle of a rectangular tricot with tops Pro(0;0), BUT(0;8), At(8,0).

Know: a) a thick layer of humor;

two-world discrete rosepodil vipadkovy

Most often, the result is described by a number of variable values: . For example, the weather in a given city at the singing hour can be characterized by such fluctuations: X 1 - temperature, X 2 - vice, X 3 - moisture content, X 4 - wind swept.

In this way we can speak about a rich variety of values ​​or about a system of values.

Let's take a look at the two-world vipad value, we can see the meaning of such a splinter of numbers. Geometrically, a two-dimensional vipadical value can be shaped like a vipadical point on a plane.

Yakscho folders Xі Y- discrete vipadic values, then - a discrete two-dimensional vipadic value, but Xі Y- uninterrupted, then - uninterrupted two-world vipadic value.

The law of divergence of differences of the two-world vipadkovy value is called the difference between the possible values ​​of those two dimensions.

The law of subdivision of a two-world discrete variable value can have tasks in a table with a sub-entrance (div. table 6.1), de X took on meaning x i, and warehouses Y- value y j .

Table 6.1.1.

Oskіlki podії, to establish the same group of pairwise insane podіy, the sum of imovirnosti dorivnyuє 1, tobto.

From tables 6.1 you can know the law of distribution of single-storey warehouses Xі Y.

butt 6.1.1 . Know the laws of the distribution of warehouses Xі Y, As a rule, a divergence of two-dimensional volatility was established in Table 6.1.2.

Table 6.1.2.

How to fix the value of one of the arguments, for example, subtracting the value X called umovny rozpodilom. In a similar way, the mental rozpodil Y.

butt 6.1.2 . According to the rozpodіlu dvovimіrnoї vipadkovoї value given tab. 6.1.2, know: a) the mental law of distribution of the warehouse X for your mind; b) the mental law has been subdivided Y mind you, what.

Solution. Cleverness of storage facilities Xі Y calculate with formulas

Umovny law rozpodіlu X for your mind, let me see

Control: .

The law of subdivision of a two-world vipadkovy magnitude can be put at a glance functions, What is meant for the skin bet of numbers X wake up meaning, less X, i at Y wake up meaning, less y:

Geometrically, the function means the possibility of trapping a vertical point into an infinite square with a vertex at the point (Fig. 6.1.1).

Significantly domineering.

• 1. Function range - , then. .
• 2. Function - an indestructible function behind the skin argument.
• 3. Mayat the place of borderline spivvіdnoshennia:

When the function of the division of the system becomes equal to the function of the division of the warehouse X, then. .

Similarly, .

Knowingly, you can know the possibility of hitting a point in the middle of the rectangle ABCD.

And myself,

Stock 6.1.3. The two-dimensional discrete vipad value is given by the table rozpodіlu

Know the function of the subdivision.

Solution. Values ​​for different discrete warehouses Xі Y be familiar with the sums of the current abilities with indexes iі j, for yaks, . Todі, yakscho i, then (podії i - not possible). Similarly, we take:

yakscho i, then;

yakscho i, then;

yakscho i, then;

yakscho i, then;

yakscho i, then;

yakscho i, then;

yakscho i, then;

yakscho i, then;

yakscho th, then.

Withdrawing the results is formalized in the table (6.1.3) value:

For two-world uninterrupted vipadkovoї value to be introduced understanding the thickness of ymovіrnostі

Geometrical width of space

The twofold scope of imovirnosti may advance power:

3. The function of the subdivision can be expressed through the formula

4. Immovability of falling of a non-stop depression value into the area of ​​the road

5. Vіdpovіdno up to power (4) functions may mіstse formulas:

Stock 6.1.4. The function of the subdivision of the two-world vertical magnitude is given

Appointment 2.7. a pair of vipadkoe numbers (X, Y), What is the point on the coordinate plane (Fig. 2.11).

Rice. 2.11.

Dvovimirna vipadkovy value - ce okremia vipadok richovimіrnoї vipadkovoї value, or vipadkovy vector.

Appointment 2.8. Vipadkovy vector value function?,(/) with terminal multiplier of possible argument t, meaning of what kind of meaning tє vipadkovoy value.

A two-dimensional vipadic value is called a non-permanent, i.e., the coordinates are non-interruptible, the other one is discrete, i.e., the coordinates are discrete.

Set the law of subdivision of two-world vipadkovyh values ​​- tse establish vіdpovіdnіst mіzh її possible values ​​\u200b\u200band imovirnіstyu tsikh value. For the methods of setting the variable values, they can be divided into bezperervnі and discrete, although є zagalnі methods of setting the law of rozpodіlu be-yakiy NE.

Discrete two-world vipad value

The discrete two-dimensional vipad value is indicated for the additional table of distributions (Table 2.1).

Table 2.1

Table rozpodіlu (spіlny rozpodіl) SV ( X, U)

Table elements are defined by the formula

The authority of the elements in the tables is subdivided:

Rozpodіl by skin coordinates is called one-dimensional or marginal:

R 1> = P(X =.d,) - marginal rozpodil CB X;

p^2) = P(Y= y,)- Marginal rozpodil SV U.

Zvyazok spilnogo rozpodіlu SV X i Y, given impersonality [p()), i = 1,..., n,j = 1,..., t(table rozpodіlu), that marginal rozpodіlu.

Similarly for SV U p-2)= X p, g

Task 2.14. Given:

Uninterrupted two-world vipadic value

/(X, y)dxdy- ymovirnosti element for dvovimіrnoї vipadkovї value (X, Y); - ymovіrnіst podadannya vypadkovoї value (X, Y) in a rectangle zі sides. cbc, dy at dx, dy -* 0:

f(x, y) - schіlnіst rozpodіlu two-dimensional vipadkovy value (X, Y). Zavdannyam / (x, y) We give the same information about roses of two-dimensional volatility.

Marginal rozpodіli are given by the offensive rank: X - schіlnistyu rozpodіlu CB X/,(x); on Y- Shchіlnistyu rozpodіlu SV U f>(y).

The task of the law of subdivision of the two-world vipadic value by the function of subdivision

In a universal way, setting the law of distribution for a discrete or non-permanent two-dimensional variable value - the function of distribution F(x, y).

Appointment 2.9. F(x, y) function- the possibility of a sleepy appearance of a sub (Hu), tobto. F(x0,y n) = = P(X y), thrown onto the coordinate plane, spent in an infinite quadrant with a vertex at the point M (x 0 y i)(U is shaded in Fig. 2.12 area).

Rice. 2.12. Illustration of the function of the subdivision F( x, y)

Domineering functions F(x, y)

• 1) 0 1;
• 2) F(-oo,-oo) = F(x,-oo) = F(-oo, y) = 0; F( oo, oo) = 1;
• 3) F(x, y)- Do not change for skin argument;
• 4) F(x, y) - uninterrupted evil from below;
• 5) convenience of roses:

F(x, X: F(x, oo) = F, (x); F(y, oo) - marginal rozpodil by Y F( oo, y) = F2(y). Zvyazok /(x, y) h F(x, y):

Zvyazok spіlnoї bushes with marginal. Dana f(x, y). Take away the marginal thickening of the rose f(x),f2(y)".

Variation of independent coordinates of a two-world vertical value

Appointment 2.10. SW Xі Independent(nz), as if they were independent, if they were under the skin, they were tied with skin from tsikh ST. Z vyznachennya NZ NE vyplivaє:

• 1 )Pij = p X) pf
• 2 ) F(x, y) = F l (x) F 2 (y).

It appears that for independent NE Xі Y viconano ta

3 ) f (x, y) = J (x) f, (y).

It is reported that for independent NE Xі Y2) 3). Bringing, a) Let vikonano 2), tobto.

at the same time F(x, y) = f J f(u,v)dudv, stars and screaming 3);

b) let me now vikonano 3), todi

tobto. Correct 2).

Let's take a look at the meeting.

Schedule 2.15. Rozpodіl inserted by the next table:

We will be marginalized:

Acceptable P(X = 3, U = 4) = 0,17 * P(X = 3) P (U \u003d 4) \u003d 0.1485 => => CB X and Uzalezhneni.

The function of the subdivision:

Task 2.16. Rozpodіl inserted by the next table:

Acceptable P tl = 0.2 0.3 = 0.06; P 12 \u003d 0.2? 0.7 = 0.14; P2l = 0,8 ? 0,3 = = 0,24; R 22 - 0.8 0.7 = 0.56 => SW Xі Y nz.

Task 2.17. Dana / (x, y) = 1/st exp| -0.5 (d "+ 2xy + 5r/2)]. Know Oh)і /Ay)-

Solution

(Dolіchit independently).

Let me give you a two-world vipadic value $ (X, Y) $.

Appointment 1

The law of subdivision of the two-dimensional variable value $(X,Y)$ is the impersonal possible pairs of numbers $(x_i,\ y_j)$ (de $x_i \epsilon X,\ y_j \epsilon Y$) and their uniformities $p_(ij)$ .

Most often, the law of distribution of the two-world variable magnitude is written as a table (Table 1).

Figure 1. The law of subdivision of a two-world variable magnitude.

Guess now the theorem about the folding of the immovability of independent subdivisions.

Theorem 1

The mobility of the sum of the final number of independent subdivisions $(\ A)_1$, $(\ A)_2$, ... , $\ (\ A)_n$ is calculated according to the formula:

Using this formula, you can take the law for the skin component and the two-dimensional volatility, so that:

It is clear that the sum of all the possibilities of a two-world system may look like this:

Let's take a look at the report (in stages) of the task, which, according to the understanding of the law, was subdivided into a two-world vipadkovy value.

butt 1

The law was subdivided into a two-world vipadkovy magnitude of tasks by the following table:

Baby 2.

To know the laws of subdivision of the magnitudes of $X,\Y$, $X+Y$ and to reconsider the totality of the totality of the individual in the skin case.

1. Let's know the number of subdivisions of the vipad value $X$. The Vipad value $X$ can be the value $x_1=2,$ $x_2=3$, $x_3=5$. For the meaning of the difference, we can use Theorem 1.

Let's know the sum of the sums of abilities $x_1$ in the following order:

Baby 3.

Similarly, we know $P\left(x_2\right)$ and $P\left(x_3\right)$:

\ \

Baby 4.

1. Now we know that we have subdivided a certain amount of $Y$. The Vipad value $Y$ can be the value $x_1=1,$ $x_2=3$, $x_3=4$. For the meaning of the difference, we can use Theorem 1.

Let's know the sum of abilities $y_1$ in the following order:

Baby 5.

Similarly, we know $P\left(y_2\right)$ and $P\left(y_3\right)$:

\ \

So, the law of distribution of the value of $X$ can look like this:

Baby 6.

Let's reconsider the vikonannya of equivalence of the total sum of emotions:

1. The law of rozpodіlu vipadkovoї value $X+Y$ has lost its significance.

Significantly її transparency in terms of $Z$: $Z=X+Y$.

On the back of the hand, we know how much the value can be given. For which pairwise summable values ​​of $X$ and $Y$. We take away the next values: 3, 4, 6, 5, 6, 8, 6, 7, 9. Now, when the values ​​have fluctuated, we take into account that the variable value $X+Y$ can take the value $z_1=3, z_2=4 \ z_3=5,\z_4=6,\z_5=7,\z_6=8,\z_7=9.\$

Known for the cob $P(z_1)$. Since the value of $ z_1 $ is single, it is known as an offensive rank:

Baby 7.

Similarly, all the properties are known, krіm $P(z_4)$:

We now know $P(z_4)$ as follows:

Baby 8.

So, the law of the distribution of the value of $Z$ can look like this:

Baby 9.

Let's reconsider the vikonannya of equivalence of the total sum of emotions:

The vipad value is called two-dimensional ( X, Y), possible values ​​of such a bet of numbers ( x, y). warehousemen Xі Y, which are looked at at once, appease system two vipadkovyh values.

A two-dimensional value can be geometrically shaped like a vipadical point M(X; Y) on the flat xOy abo yak vipadkovy vector OM.

Discrete call a two-world value, warehouses that are discrete.

Uninterrupted naming a two-dimensional value, warehouses that are uninterrupted.

Law rozpodіlu ymovirnosti dvovimіrnoї vypadkovoї value called vіdpovіdnіst mіzh possible values ​​їkh ymovіrnosti.

The law of subdivision of a discrete two-world variable value can have tasks: a) look at the table with a sub-entrance, which value can be avenged; b) analytically, for example, looking at the functions of the subdivided.

The function was subdivided univariances of a two-dimensional falling value is called a function F(x, y), which is used for skin betting numbers (x, y) imovirnіst that X accept a value less than x, і with tsmu Y wake up meaning, less y:

F(x, y) = P(X< x, Y < y).

Geometrically, qiu equanimity can be visualized like this: F(x, y)є ymovirnist that vipadkova point ( X, Y) eat at a non-sinking quadrant with a vertex ( x,y) rotting levoruch and lower tsієї peaks.

In other words, replace the term "function of the subdivided" and win the term "integral function".

The function of rozpodіlu may have the same power:

Power 1. The value of the function rozpodіlu satisfies the underlying unevenness

0 ≤ F (x, y) ≤ 1.

Power 2. The function was subdivided into an immutable function behind the skin argument:

F(x 2 , y) ≥ F(x 1 , y), so x 2 > x 1 ,

F(x, y 2) ≥ F(x, y 1), so y 2 > y 1 .

Power 3. Mayut the place of borderline spіvvіdshennya:

1) F(–∞, y) = 0,

3) F(–∞, –∞) = 0,

2) F(x, –∞) = 0,

4) F(∞, ∞) = 1.

Power 4. a) At=∞ the function of distribution subdivision of the system becomes the function of distribution subdivision of warehouse X:

F(x, ∞) = F 1 (x).

b) For x = ∞ the function of the distribution subdivision of the system becomes the function of the distribution subdivision of the warehouse Y:

F(∞, y) = F 2 (y).

Vykoristovuyu function rose, you can know the possibility of falling vpadkovy point in a rectangle x 1< X < x 2 , y 1 < Y < у 2 :

P(x1< X < x 2 , y 1 < Y < у 2) = – .

Shіlnistyu splіl rozpodіl ymovіrnosti (two-world thick ymovіrnostі) uninterrupted two-world volatility value is called another zm_shanu similar to the function of rozpodіlu:

Another substitute for the term "two-world denseness of imovirnosti" and the term "differential function of the system".

The width of the folded rosette can be seen as between the difference between the degree of falling of the drop point at the rectangle from the sides D x and D y up to the square of the rectangle, if offending the sides of the pragmatic zero; geometrically її it is possible to vitlumachiti as a surface, as a name the surface rose under.

Knowing the thickness of the subdivision, you can know the function of the subdivision for the formula

The mobility of the vertical drop point (X, Y) in the area D varies

The twofold scope of imovirnosti may advance power:

Power 1. Dvovimіrna shіlnіst ymovіrnostі nevid'єmna:

f(x,y) ≥ 0.

Power 2. Suspended unmatched integral with indistinct boundaries in the form of a two-world gap and a healthy one:

Sokrema, since all possible values ​​(X, Y) lie at the end of D, then

226. Given the distribution of the values ​​of a discrete two-dimensional variable value:

Know the laws of the distribution of warehouses.

228. Given the function of a subdivision of a two-world vertical magnitude

Know the ability to hit the drop point ( X, Y x = 0, x= p/4, y= p/6, y= p/3.

229. Know the ability of hitting a point of fall ( X, Y) near a rectangle, surroundings with straight lines x = 1, x = 2, y = 3, y= 5

230. Given a function of a subdivision of a two-world vertical magnitude

Know the double thickness of the system's flexibility.

231. When x 2 + y 2 ≤ R 2 two-world scope of imovirnosti; colo pose f(x, y)= 0. Know: a) fast C; b) the ability to hit the drop point ( X, Y) near the radius r= 1 centered on the cob of coordinates, so R = 2.

232. The first quadrant has the function of subdivision of the system of two variances F(x, y) = 1 + 2 - x - 2 - y + 2 - x-y. To know: a) the double thickness of the system's flexibility; b) the ability to hit the drop point ( X, Y) at trikutnik with vertices A(1; 3), B(3; 3), C(2; 8).

8.2. Umovnі zakoni rozpodіla ymovіrnosti warehousing
a discrete two-world drop-down value

Bring warehouse Xі Y discrete and possibly so possible values: x 1, x 2, …, x n; y 1 , y 2 , …, y m.

Warehouse manager X at Y=y j(j takes the same value for all possible values ​​of X)

p(x 1 | y j), p (x 2 | y j), …, p (x n | y j).

In a similar way, the mental rozpodil Y.

Mindfulness of warehouse X and Y are calculated according to formulas

For control, calculate the number of points to change, so that the sum of the intelligences of the mental one was divided into more than one.

233. A discrete two-world vipad value is given ( X, Y):

Know: a) the mental law has been subdivided X mind what Y=10; b) the mental law has been subdivided Y mind what X=6.

8.3. The knowledge of the powers and the intellectual laws of the rozpodіlu
storage uninterrupted two-world drop size

Shіlnіst raspodіlu odnієї zі warehousing dorіvnyuє to the implicit integral with inconspicuous boundaries in the thicket of the sleeper rose under the system, moreover, the integration of the other warehouses is changed:

Here it is said that it is possible for the skin values ​​of the warehouses to lie down to the entire numerical axis; if it is possible for the values ​​to lie on the end interval, then the interintegration takes the last end numbers.

Umovnoy schіlnistyu rozpodіl warehousing X at a given value Y=y call the extension of the width of the sleeper substructure of the system to the thickness of the warehouse substructure Y:

In a similar way, the mental capacity of the distribution of the warehouse Y:

Yakshcho umovnі schіlnostі rozpodіl vypadkovymi values Xі Y equal to their insane powers, such magnitudes are independent.

Rivnomirnym call the rozpodіl dvovimіrnoї uninterrupted volatility value ( X, Y), even though in the area, which must lie with the possible values ​​( x, y), the thickness of a sleeping rose under the ymovіrnosti zberіgaє postіyne znachennya.

235. Given the width of a sleepy rose under a non-permanent two-dimensional vertical dimension (X, Y)

Know: a) schіlnіst rozpodіlu warehousing; b) smart ass rozpodіlu what to establish.

236. The thickness of a sleepy rose under a non-permanent two-dimensional volatility ( X, Y)

Know: a) constant multiplier C; b) bushy rozpodіlu warehousing; c) wiser gustini rozpodіlu warehousing.

237. Uninterrupted two-world vipad value ( X, Y) is divided evenly in the middle of the rectangle with the center of symmetry on the cob of coordinates and sides 2a and 2b, parallel to the coordinate axes. To know: a) the double thickness of the system's flexibility; b) bushes rose under the warehouse.

238. Uninterrupted two-world vipad value ( X, Y) evenly split in the middle of a rectangular tricot with tops O(0; 0), BUT(0; 8), At(8; 0). To know: a) the double thickness of the system's flexibility; b) the width and the intelligence width of the warehouses.

8.4. Numerical characteristics of an uninterrupted system
two vipadic magnitudes

Knowing the width of the distribution of the warehouse X and Y of the uninterrupted two-world drop-down value (X, Y), you can know their mathematical dispersion:

Sometimes it’s better to victorious formulas to avenge the two-world thicket of fluency ( underlying integrals are taken according to the area of ​​possible values ​​of the system):

Cob moment n k, s order k+s system ( X, Y) to name mathematically the creation X k Y s:

nk, s = M.

Zokrema,

n 1.0 = M(X), n 0.1 = M(Y).

Central moment m k, s order k+s system ( X, Y) to name mathematically ochіkuvannya creative work k th ta s-th step:

m k, s = M (k ∙ s).

Zokrema,

m 1.0 = M = 0, m 0.1 = M = 0;

m 2.0 = M 2 = D (X), m 0.2 = M 2 = D (Y);

Correlation moment m xу system ( X, Y) call the central moment m 1.1 order 1 + 1:

m xу = M( ∙ ).

Correlation coefficient the values ​​of X and Y are called the correlation moment to the creation of the root-mean-square deviations of these values:

r xy = m xy / (s x s y).

The coefficient of correlation is a bezrozmіrna value, moreover | rxy| ≤ 1. Correlation coefficient to serve to assess the accuracy of the linear connection between Xі Y: the closer the absolute value of the correlation coefficient to one, the stronger the link; The closer the absolute value of the correlation coefficient is to zero, then the connection is weaker.

Correlations name two reversible magnitudes, which is to say that their correlation moment is equal to zero.

Uncorrelated name two reversible values, so that their correlation moment is equal to zero.

Two correlated values ​​are also deposits; Although there are two values ​​of deposits, all stinks can be both correlated and uncorrelated. Due to the independence of the two values, the uncorrelatedness is evident, but due to the noncorrelation, it is still not possible to make vysnovkas about the independence of these values ​​(for normally separated values, the noncorrelation of these values ​​is evident).

For uninterrupted quantities X and Y correlation moment can be found using the formulas:

239. The width of a sleepy rose under a non-permanent two-dimensional vertical drop value (X, Y) is given:

Know: a) mathematical education; b) dispersion of warehouses X and Y.

240. The width of a sleepy rose under a non-permanent two-dimensional vertical drop value (X, Y) is given:

Know mathematical ochіkuvannya and variance of warehouses.

241. The width of a sleepy rose under a non-permanent two-dimensional vertical drop is given ( X, Y): f(x, y) = 2 cosx cozy square 0 ≤ x≤p/4, 0 ≤ y≤p/4; square pose f(x, y)= 0. Know the mathematical classification of warehouses.

242. To bring, that is a two-dimensional thickening of the ymovirnosti of the system of variable values ​​( X, Y) you can look at two functions, one of which can be found only in x, and insha - only in y, then the quantities Xі Y independent.

243. To bring, scho yakscho Xі Y connected with linear fallow Y = aX + b, The absolute value of the coefficient of correlation of the most important unit.

Solution. For the purpose of the correlation coefficient,

r xy = m xy / (s x s y).

m xу = M( ∙ ). (*)

We know mathematically Y:

M(Y) = M = aM(X) + b. (**)

Substituting (**) at (*), after elementary transformation taken

m xy \u003d aM 2 \u003d aD (X) \u003d as 2 x.

Vrakhovuyuchi sho

Y – M(Y) = (aX + b) – (aM(X) + b) = a,

we know dispersion Y:

D(Y) = M 2 = a 2 M 2 = a 2 s 2 x.

Zvіdsi s y = |a|s x. Also, correlation coefficient

Yakscho a> 0, then rxy= 1; yakscho a < 0, то rxy = –1.

Otzhe, | rxy| = 1, which was necessary to bring.