Local de Moivre-Laplace theorem. 0 and 1, then the probability P t p of that, that the event A will occur m times in n independent trials for a sufficiently large number n, is approximately equal to
- Gaussian function and
The larger and, the more accurate the approximate formula (2.7), called by the local Moivre-Laplace formula. Approximate probabilities R TPU given by the local formula (2.7) are used in practice as exact ones for pru of the order of two or more tens, i.e. given that pru > 20.
To simplify the calculations associated with the use of formula (2.7), a table of values of the function /(x) has been compiled (Table I, given in the appendices). When using this table, it is necessary to keep in mind the obvious properties of the function f(x) (2.8).
- 1. Function/(X) is even, i.e. /(-x) = /(x).
- 2. Function/(X) - monotonically decreasing for positive values X, and at x -> co /(x) -» 0.
- (In practice, we can assume that even for x > 4 /(x) « 0.)
[> Example 2.5. In some area, out of every 100 families, 80 have refrigerators. Find the probability that out of 400 families 300 have refrigerators.
Decision. The probability that a family has a refrigerator is p = 80/100 = 0.8. As P= 100 is large enough (condition pru= = 100 0.8(1-0.8) = 64 > 20 satisfied), then we apply the local Moivre-Laplace formula.
First, we define by formula (2.9)
Then by formula (2.7)
(the value /(2.50) was found from Table I of the appendices). The rather small value of the probability /300,400 should not be in doubt, since apart from the event
“exactly 300 families out of 400 have refrigerators” 400 more events are possible: “0 out of 400”, “1 out of 400”,..., “400 out of 400” with their own probabilities. Together, these events form a complete group, which means that the sum of their probabilities is equal to one. ?
Let, in the conditions of Example 2.5, it is necessary to find the probability that from 300 to 360 families (inclusive) have refrigerators. In this case, according to the addition theorem, the probability of the desired event
In principle, each term can be calculated using the local Moivre-Laplace formula, but a large number of terms makes the calculation very cumbersome. In such cases, the following theorem is used.
Integral theorem of Moivre - Laplace. If the probability p of the occurrence of event A in each trial is constant and different from 0 and 1, then the probability of, that the number m of the occurrence of event A in n independent trials lies between a and b (inclusive), for a sufficiently large number n is approximately equal to
- function(or integral of probabilities) Laplace",
(The proof of the theorem is given in Section 6.5.)
Formula (2.10) is called Moivre-Laplace integral formula. The more P, the more accurate the formula. When the condition pru > > 20 the integral formula (2.10), as well as the local one, gives, as a rule, an error in calculating probabilities that is satisfactory for practice.
The function Φ(dg) is tabulated (see Table II of the appendices). To use this table, you need to know the properties of the function Ф(х).
1. Function f(x) odd, those. F(-x) = -F(x).
?
Shall we change the variable? = -G. Then (k =
= -(12. The limits of integration for variable 2 will be 0 and X. Get
since the value of the definite integral does not depend on the designation of the integration variable. ?
2. The function Ф(х) is monotonically increasing, and for x ->+co f(.g) -> 1 (in practice, we can assume that already at x > 4 φ(x)~ 1).
Since the derivative of the integral with respect to the variable upper limit is equal to the integrand at the value of the upper limit, r.s.
, and is always positive, then Ф(х) increases monotonically
along the whole number line.
We make a change of variable, then the limits of integration do not change and
(since the integral of an even function
Given that 
(Euler integral - Poisson), we get
?
O Example 2.6. Using the data of Example 2.5, calculate the probability that from 300 to 360 (inclusive) families out of 400 have refrigerators.
Decision. We apply the integral theorem of Moivre - Laplace (pr= 64 > 20). First, we define by formulas (2.12)
Now, according to formula (2.10), taking into account the properties of Ф(.т), we obtain
(according to Table II of appendices?
Consider a consequence of the integral theorem of Moivre - Laplace. Consequence. If the probability p of the occurrence of event A in each trial is constant and different from 0 and I, then for a sufficiently large number n of independent trials, the probability that:
a) the number m of occurrences of the event A differs from the product pr by no more than e > 0 (in absolute value), those.
b) the frequency of the t / n event A lies within from a to r ( including- respectfully, i.e.
in) the frequency of event A differs from its probability p by no more than A > 0 (in absolute value), i.e.
A) Inequality |/?7-7?/?| is equivalent to a double inequality pr-e Therefore, by the integral formula (2.10)
- b) Inequality and is equivalent to the inequality and at a = pa and b= /?r. Replacing in formulas (2.10), (2.12) the quantities a and b obtained expressions, we obtain the provable formulas (2.14) and (2.15).
- c) Inequality mjn-p is equivalent to the inequality t-pr Replacing in the formula (2.13) r = Ap, we obtain the formula (2.16) to be proved. ?
[> Example 2.7. Using the data in Example 2.5, calculate the probability that 280 to 360 families out of 400 have refrigerators.
Decision. Calculate the probability Р 400 (280 t pr \u003d 320. Then according to the formula (2.13)
[> Example 2.8. According to statistics, on average, 87% of newborns live to be 50 years old.
- 1. Find the probability that out of 1000 newborns the proportion (frequency) of those who survived to 50 years of age will: a) be within the range from 0.9 to 0.95; b) will differ from the probability of this event by no more than 0.04 (but in absolute value).
- 2. At what number of newborns with a reliability of 0.95 will the proportion of those who survived to 50 years of age be within the limits from 0.86 to 0.88?
Decision. 1a) Probability R that a newborn will live to 50 years is 0.87. As P= 1000 large (condition prd=1000 0.87 0.13 = 113.1 > 20 satisfied), then we use the corollary of the integral theorem of Moivre - Laplace. First, we define by the formulas (2.15)
Now according to the formula (2.14)
1, b) By formula (2.16)
Because inequality 
is equivalent to the inequality
the result obtained means that it is practically certain that from 0.83 to 0.91 of the number of newborns out of 1000 will live to 50 years. ?
2. By condition 
or
According to the formula (2.16)
at A = 0.01 
According to the table II applications F(G) = 0.95 at G = 1.96, therefore,
where 
those. condition (*) can be guaranteed with a significant increase in the number of considered newborns up to P = 4345. ?
- The proof of the theorem is given in Section 6.5. The probabilistic meaning of the quantities pr, prs( is established in paragraph 4.1 (see note on p. 130).
- The probabilistic meaning of the value pf/n is established in paragraph 4.1.
the pressure directly under the convex surface of the liquid is greater than the pressure under the flat surface of the liquid, and the pressure under the concave surface of the liquid is less than the pressure under the flat surface.
Calculation of pressure under the spherical surface of a liquid
It is a thin layer of water, which has two bounding surfaces: internal and external. The radii of curvature of these surfaces can be considered the same, since the film thickness is thousands of times smaller than the bubble radius. Water from this layer gradually drains, the layer becomes thinner and finally breaks. So the bubbles do not float on the water for very long: from fractions of a second to ten seconds. It should be noted that as the water film becomes thinner, the bubble size practically does not change.
Let us calculate the excess pressure in such a bubble. For simplicity, consider a single-layer hemisphere of radius r, located on a horizontal surface, we will also assume that there is no air outside. The film is held on the shaded surface due to wetting (Fig. 2.3). In this case, along the boundary of contact with the surface, a surface tension force equal to
where is the coefficient of surface tension of the liquid,
The length of the film-surface interface is equal to .
That is, we have:
.
This force acting on the film, and through it on the air, is directed perpendicular to the surface (see Fig. 2.3). So the air pressure on the surface and therefore inside the bubble can be calculated as follows:
Where F is the surface tension force equal to,
S - surface area: .
Substituting the value of the force F and the area S in the formula for calculating the pressure, we get:
and finally.
In our example with an air bubble on the surface of the water, the film is double and, therefore, the excess pressure is .
Figure 2.4 shows examples of single-layer spherical surfaces that can form on the surface of a liquid. Above the liquid is a gas that has pressure.
Capillarity (from Latin capillaris - hair), capillary effect - a physical phenomenon consisting in the ability of liquids to change the level in tubes, narrow channels of arbitrary shape, porous bodies. The rise of the liquid occurs when the channels are wetted with liquids, for example, water in glass tubes, sand, soil, etc. The decrease in liquid occurs in tubes and channels that are not wetted by liquid, for example, mercury in a glass tube.
On the basis of capillarity, the vital activity of animals and plants, chemical technologies, and everyday phenomena are based (for example, rising kerosene along the wick in a kerosene lamp, wiping hands with a towel). Soil capillarity is determined by the rate at which water rises in the soil and depends on the size of the gaps between soil particles.
Laplace formula
Consider a thin liquid film whose thickness can be neglected. In an effort to minimize its free energy, the film creates a pressure difference from different sides. This explains the existence of soap bubbles: the film is compressed until the pressure inside the bubble exceeds the atmospheric pressure by the value of the additional pressure of the film. The additional pressure at a point on the surface depends on the average curvature at that point and is given by the Laplace formula:
Here R 1,2 are the radii of the main curvatures at a point. They have the same sign if the corresponding centers of curvature lie on the same side of the tangent plane at the point, and they have a different sign if they lie on the opposite side. For example, for a sphere, the centers of curvature at any point on the surface coincide with the center of the sphere, so
For the case of the surface of a circular cylinder of radius R, we have
It is known that the surface of the liquid near the walls of the vessel is curved. The free surface of a liquid curved near the walls of the vessel is called the meniscus.(Fig. 145).
Consider a thin liquid film whose thickness can be neglected. In an effort to minimize its free energy, the film creates a pressure difference from different sides. Due to the action of surface tension forces in liquid droplets and inside soap bubbles, additional pressure(the film is compressed until the pressure inside the bubble does not exceed the atmospheric pressure by the value of the additional pressure of the film).
 Rice. 146. |
Consider the surface of a liquid resting on some flat contour (Fig. 146, a). If the surface of the liquid is not flat, then its tendency to contract will lead to the appearance of pressure, additional to that experienced by a liquid with a flat surface. In the case of a convex surface, this additional pressure is positive (Fig. 146, b), in the case of a concave surface - negatively (Fig. 146, in). In the latter case, the surface layer, seeking to contract, stretches the liquid.
The magnitude of the additional pressure, obviously, should increase with an increase in the coefficient of surface tension and surface curvature .
 Rice. 147. |
.
This force presses both hemispheres to each other along the surface and, therefore, causes additional pressure: 
The curvature of a spherical surface is the same everywhere and is determined by the radius of the sphere. Obviously, the smaller , the greater the curvature of the spherical surface.
The excess pressure inside the soap bubble is twice as much, since the film has two surfaces:
Additional pressure causes a change in the liquid level in narrow tubes (capillaries), as a result of which it is sometimes called capillary pressure.
The curvature of an arbitrary surface is usually characterized by the so-called average curvature, which may be different for different points on the surface.
The value gives the curvature of the sphere. In geometry, it is proved that the half-sum of the reciprocal radii of curvature for any pair of mutually perpendicular normal sections has the same value:
. (1)
This value is the average curvature of the surface at a given point. In this formula, the radii are algebraic quantities. If the center of curvature of a normal section is below a given surface, the corresponding radius of curvature is positive; if the center of curvature lies above the surface, the radius of curvature is negative (Fig. 148).
 Rice. 148. |
For example, for a sphere, the centers of curvature at any point on the surface coincide with the center of the sphere, and therefore . For the case of the surface of a circular cylinder of radius, we have: , and .
It can be proved that for a surface of any shape the relation is true:
Substituting expression (1) into formula (2), we obtain the formula for additional pressure under an arbitrary surface, called Laplace formula(Fig. 148):
. (3)
The radii and in formula (3) are algebraic quantities. If the center of curvature of a normal section is below a given surface, the corresponding radius of curvature is positive; if the center of curvature lies above the surface, the radius of curvature is negative.
Example. If there is a gas bubble in the liquid, then the surface of the bubble, trying to shrink, will exert additional pressure on the gas .
Let us find the radius of a bubble in water at which the additional pressure is 1 atm. .Coefficient of surface tension of water at equal 
. Therefore, for the following value is obtained: .
For sufficiently large, the Bernoulli formula gives cumbersome calculations. Therefore, in such cases, the local Laplace theorem is used.
Theorem(local Laplace theorem). If the probability p of the occurrence of event A in each trial is constant and different from 0 and 1, then the probability 
the fact that event A will appear exactly k times in n independent trials is approximately equal to the value of the function:
,
.
There are tables that contain the values of the function 
, for positive values of x.
Note that the function 
even.
So, the probability that event A will appear exactly k times in n trials is approximately equal to
, where 
.
Example. 1500 seeds were sown on the experimental field. Find the probability that seedlings will produce 1200 seeds if the probability that a seed will germinate is 0.9.
Decision. 
Laplace integral theorem
The probability that in n independent trials event A will occur at least k1 times and at most k2 times is calculated by Laplace's integral theorem.
Theorem(Laplace integral theorem). If the probability p of the occurrence of event a in each trial is constant and different from 0 and 1, then the probability that event A in n trials will appear at least k 1 times and at most k 2 times is approximately equal to the value of a certain integral:
.
Function 
is called the Laplace integral function, it is odd and its value is found in the table for positive values of x.
Example. In the laboratory, from a batch of seeds with a germination rate of 90%, 600 seeds were sown, which sprouted, not less than 520 and not more than 570.
Decision.
Poisson formula
Let n independent trials be performed, the probability of occurrence of event A in each trial is constant and equal to p. As we have already said, the probability of the occurrence of event A in n independent trials exactly k times can be found using the Bernoulli formula. For sufficiently large n, the local Laplace theorem is used. However, this formula is unsuitable when the probability of an event occurring in each trial is small or close to 1. And when p=0 or p=1, it is not applicable at all. In such cases, the Poisson theorem is used.
Theorem(Poisson's theorem). If the probability p of the occurrence of event A in each trial is constant and close to 0 or 1, and the number of trials is sufficiently large, then the probability that in n independent trials event A will occur exactly k times is found by the formula:
.
Example. The 1,000-page typewritten manuscript contains 1,000 typographical errors. Find the probability that a randomly selected page contains at least one misprint.
Decision.
Questions for self-test
Formulate the classical definition of the probability of an event.
Formulate theorems of addition and multiplication of probabilities.
Define a complete group of events.
Write down the formula for the total probability.
Write down the Bayes formula.
Write down the Bernoulli formula.
Write down Poisson's formula.
Write down the local Laplace formula.
Write down Laplace's integral formula.
Topic 13. Random variable and its numerical characteristics
Literature: ,,,,,.
One of the basic concepts in probability theory is the concept of a random variable. So it is customary to call a variable that takes on its values depending on the case. There are two types of random variables: discrete and continuous. Random variables are usually denoted X,Y,Z.
A random variable X is called continuous (discrete) if it can take only a finite or countable number of values. A discrete random variable X is defined if all its possible values x 1 , x 2 , x 3 ,…x n are given (the number of which can be either finite or infinite) and the corresponding probabilities p 1 , p 2 , p 3 ,… p n.
The distribution law of a discrete random variable X is usually given by the table:
The first line contains the possible values of the random variable X, and the second line contains the probabilities of these values. The sum of the probabilities with which the random variable X takes on all its values is equal to one, that is
p 1 + p 2 + p 3 + ... + p n \u003d 1.
The distribution law of a discrete random variable X can be represented graphically. To do this, points M 1 (x 1, p 1), M 2 (x 2, p 2), M 3 (x 3, p 3), ... M n (x n, p n) are built in a rectangular coordinate system and connect them with segments direct. The resulting figure is called the distribution polygon of the random variable X.
Example. The discrete value X is given by the following distribution law:
It is required to calculate: a) mathematical expectation M(X), b) variance D(X), c) standard deviation σ.
Decision . a) The mathematical expectation of M(X), a discrete random variable X is the sum of pairwise products of all possible values of the random variable and the corresponding probabilities of these possible values. If a discrete random variable X is given using the table (1), then the mathematical expectation M(X) is calculated by the formula
М(Х)=х 1 ∙р 1 +х 2 ∙р 2 +х 3 ∙р 3 +…+х n ∙p n . (2)
The mathematical expectation M(X) is also called the average value of the random variable X. Applying (2), we get:
М(Х)=48∙0.2+53∙0.4+57∙0.3 +61∙0.1=54.
b) If M(X) is the expectation of a random variable X, then the difference X-M(X) is called deviation random variable X from the average value. This difference characterizes the scattering of a random variable.
dispersion(scattering) of a discrete random variable X is the mathematical expectation (mean value) of the squared deviation of a random variable from its mathematical expectation. Thus, by definition, we have:
D(X)=M 2 . (3)
We calculate all possible values of the square of the deviation.
2 =(48-54) 2 =36
2 =(53-54) 2 =1
2 =(57-54) 2 =9
2 =(61-54) 2 =49
To calculate the variance D(X), we compose the distribution law of the squared deviation and then apply formula (2).
D(X)= 36∙0.2+1∙0.4+9∙0.3 +49∙0.1=15.2.
It should be noted that the following property is often used to calculate the variance: the variance D(X) is equal to the difference between the mathematical expectation of the square of the random variable X and the square of its mathematical expectation, that is
D(X)-M(X 2)- 2 . (4)
To calculate the variance using formula (4), we compose the distribution law of the random variable X 2:
Now let's find the mathematical expectation M(X 2).
М(Х 2)= (48) 2 ∙0.2+(53) 2 ∙0.4+(57) 2 ∙0.3 +(61) 2 ∙0.1=
460,8+1123,6+974,7+372,1=2931,2.
Applying (4), we get:
D(X)=2931.2-(54) 2=2931.2-2916=15.2.
As you can see, we got the same result.
c) The dimension of the variance is equal to the square of the dimension of the random variable. Therefore, to characterize the dispersion of possible values of a random variable around its mean value, it is more convenient to consider a value that is equal to the arithmetic value of the square root of the variance, that is 
. This value is called the standard deviation of the random variable X and denoted by σ. Thus
σ= 
. (5)
Applying (5), we have: σ= 
.
Example. The random variable X is distributed according to the normal law. Mathematical expectation М(Х)=5; variance D(X)=0.64. Find the probability that, as a result of the test, X will take a value in the interval (4; 7).
Decision.It is known that if a random variable X is given by a differential function f(x), then the probability that X takes a value belonging to the interval (α,β) is calculated by the formula
. (1)
If the value X is distributed according to the normal law, then the differential function
,
where a=M(X) and σ= 
. In this case, we obtain from (1)
. (2)
Formula (2) can be transformed using the Laplace function.
Let's make a substitution. Let be 
. Then 
or dx=σ∙
dt.
Hence 
, where t 1 and t 2 are the corresponding limits for the variable t.
Reducing by σ, we have
From the input substitution 
follows that 
and 
.
Thus,
(3)
According to the condition of the problem, we have: a=5; σ= 
=0.8; α=4; β=7. Substituting these data into (3), we get:
=F(2.5)-F(-1.25)=
\u003d F (2.5) + F (1.25) \u003d 0.4938 + 0.3944 \u003d 0.8882.
Example. It is believed that the deviation of the length of manufactured parts from the standard is a random variable distributed according to the normal law. Standard length (expectation) a = 40 cm, standard deviation σ = 0.4 cm. Find the probability that the deviation of the length from the standard will be no more than 0.6 cm in absolute value.
Decision.If X is the length of the part, then according to the condition of the problem, this value should be in the interval (a-δ, a + δ), where a=40 and δ=0.6.
Putting in formula (3) α= a-δ and β= a+δ, we obtain
. (4)
Substituting the available data into (4), we obtain:
Therefore, the probability that the length of the manufactured parts will be in the range from 39.4 to 40.6 cm is 0.8664.
Example. The diameter of the parts manufactured by the plant is a random variable distributed according to the normal law. Standard Diameter Length a=2.5 cm, standard deviation σ=0.01. Within what limits can one practically guarantee the length of the diameter of this part, if an event with a probability of 0.9973 is taken as a reliable one?
Decision. By the condition of the problem, we have:
a=2.5; σ=0.01; .
Applying formula (4), we obtain the equality:
or 
.
According to table 2, we find that the Laplace function has such a value at x=3. Hence, 
; whence σ=0.03.
Thus, it can be guaranteed that the length of the diameter will vary between 2.47 and 2.53 cm.
Consider the surface of a liquid resting on some flat contour. If the surface of the liquid is not flat, then its tendency to contract will lead to the appearance of pressure, additional to that experienced by a liquid with a flat surface. In the case of a convex surface, this additional pressure is positive; in the case of a concave surface, it is negative. In the latter case, the surface layer, seeking to contract, stretches the liquid. Work as a teacher of the course HR records management Moscow.
The magnitude of the additional pressure, obviously, should increase with an increase in the surface tension coefficient α and surface curvature. Let us calculate the additional pressure for the spherical surface of the liquid. To do this, we cut a spherical liquid drop by a diametral plane into two hemispheres (Fig. 5).
Cross section of a spherical liquid drop.
Due to surface tension, both hemispheres are attracted to each other with a force equal to:
This force presses both hemispheres to each other along the surface S=πR2 and, therefore, causes additional pressure:
∆p=F/S=(2πRα)/ πR2=2α/R (4)
The curvature of a spherical surface is the same everywhere and is determined by the radius of the sphere R. Obviously, the smaller R, the greater the curvature of the spherical surface. The curvature of an arbitrary surface is usually characterized by the so-called average curvature, which may be different for different points on the surface.
The average curvature is determined through the curvature of the normal sections. The normal section of a surface at some point is the line of intersection of this surface with a plane passing through the normal to the surface at the point under consideration. For a sphere, any normal section is a circle of radius R (R is the radius of the sphere). The value H=1/R gives the curvature of the sphere. In general, different sections drawn through the same point have different curvatures. In geometry, it is proved that the half-sum of the reciprocal radii of curvature
H=0.5(1/R1+1/R2) (5)
for any pair of mutually perpendicular normal sections has the same value. This value is the average curvature of the surface at a given point.
The radii R1 and R2 in formula (5) are algebraic quantities. If the center of curvature of a normal section is below the given surface, the corresponding radius of curvature is positive, if the center of curvature lies above the surface, the radius of curvature is negative.
For sphere R1=R2=R, so according to (5) H=1/R. Replacing 1/R through H in (4), we get that
Laplace proved that formula (6) is valid for a surface of any shape, if by H we mean the average curvature of the surface at this point, under which the additional pressure is determined. Substituting expression (5) for the average curvature into (6), we obtain the formula for the additional pressure under an arbitrary surface:
∆p=α(1/R1+1/R2) (7)
It is called the Laplace formula.
Additional pressure (7) causes a change in the liquid level in the capillary, as a result of which it is sometimes called capillary pressure.
The existence of the contact angle leads to the curvature of the liquid surface near the walls of the vessel. In a capillary or in a narrow gap between two walls, the entire surface is curved. If the liquid wets the walls, the surface has a concave shape; if it does not wet, it is convex (Fig. 4). Such curved liquid surfaces are called menisci.
If the capillary is immersed with one end into a liquid poured into a wide vessel, then under the curved surface in the capillary the pressure will differ from the pressure along the flat surface in the wide vessel by the value ∆p defined by formula (7). As a result, when the capillary is wetted, the liquid level in it will be higher than in the vessel, and when not wetted, it will be lower.
