In probability theory and in many of its applications, various numerical characteristics of random variables are of great importance. The main ones are mathematical expectation and variance.
1. Mathematical expectation of a random variable and its properties.
Let's first consider the following example. Let the plant receive a batch consisting of N bearings. Wherein:
m 1 x 1,
m 2- number of bearings with outer diameter x 2,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
m n- number of bearings with outer diameter x n,
Here m 1 +m 2 +...+m n =N. Let's find the arithmetic mean x avg outer diameter of the bearing. Obviously,
The outer diameter of a bearing taken out at random can be considered as a random variable taking values x 1, x 2, ..., x n, with corresponding probabilities p 1 =m 1 /N, p 2 =m 2 /N, ..., p n =m n /N, since the probability p i appearance of a bearing with an outer diameter x i equal to m i /N. Thus, the arithmetic mean x avg The outer diameter of the bearing can be determined using the relation 
Let be a discrete random variable with a given probability distribution law
Values
x 1
x 2
. . .
x n
Probabilities
p 1
p 2
. . .
p n
Mathematical expectation discrete random variable is the sum of paired products of all possible values of a random variable by their corresponding probabilities, i.e. *
In this case, it is assumed that the improper integral on the right side of equality (40) exists.
Let's consider the properties of mathematical expectation. In this case, we will limit ourselves to the proof of only the first two properties, which we will carry out for discrete random variables.
1°. The mathematical expectation of the constant C is equal to this constant.
Proof. Constant C can be thought of as a random variable that can only take one value C with probability equal to one. That's why 
2°. The constant factor can be taken beyond the sign of the mathematical expectation, i.e. 
Proof. Using relation (39), we have 
3°. The mathematical expectation of the sum of several random variables is equal to the sum of the mathematical expectations of these variables:
The distribution law fully characterizes the random variable. However, often the distribution law is unknown and one has to limit oneself to less information. Sometimes it is even more profitable to use numbers that describe a random variable in total; such numbers are called numerical characteristics random variable. One of the important numerical characteristics is the mathematical expectation.
The mathematical expectation, as will be shown below, is approximately equal to the average value of the random variable. To solve many problems, it is enough to know the mathematical expectation. For example, if it is known that the mathematical expectation of the number of points scored by the first shooter is greater than that of the second, then the first shooter, on average, scores more points than the second, and, therefore, shoots better than the second.
Definition4.1: Mathematical expectation A discrete random variable is the sum of the products of all its possible values and their probabilities.
Let the random variable X can only take values x 1, x 2, … x n, whose probabilities are respectively equal p 1, p 2, … p n. Then the mathematical expectation M(X) random variable X is determined by equality
M (X) = x 1 p 1 + x 2 p 2 + …+ x n p n .
If a discrete random variable X takes a countable set of possible values, then
,
Moreover, the mathematical expectation exists if the series on the right side of the equality converges absolutely.
Example. Find the mathematical expectation of the number of occurrences of an event A in one trial, if the probability of the event A equal to p.
Solution: Random value X– number of occurrences of the event A has a Bernoulli distribution, so
Thus, the mathematical expectation of the number of occurrences of an event in one trial is equal to the probability of this event.
Probabilistic meaning of mathematical expectation
Let it be produced n tests in which the random variable X accepted m 1 times value x 1, m 2 times value x 2 ,…, m k times value x k, and m 1 + m 2 + …+ m k = n. Then the sum of all values taken X, is equal x 1 m 1 + x 2 m 2 + …+ x k m k .
The arithmetic mean of all values taken by the random variable will be
Attitude m i/n- relative frequency W i values x i approximately equal to the probability of the event occurring p i, Where , That's why
The probabilistic meaning of the result obtained is as follows: mathematical expectation is approximately equal(the more accurate, the greater the number of tests) arithmetic mean of observed values of a random variable.
Properties of mathematical expectation
Property1:The mathematical expectation of a constant value is equal to the constant itself
Property2:The constant factor can be taken beyond the sign of the mathematical expectation
Definition4.2: Two random variables are called independent, if the distribution law of one of them does not depend on what possible values the other quantity took. Otherwise random variables are dependent.
Definition4.3: Several random variables called mutually independent, if the laws of distribution of any number of them do not depend on what possible values the other quantities took.
Property3:The mathematical expectation of the product of two independent random variables is equal to the product of their mathematical expectations.
Consequence:The mathematical expectation of the product of several mutually independent random variables is equal to the product of their mathematical expectations.
Property4:The mathematical expectation of the sum of two random variables is equal to the sum of their mathematical expectations.
Consequence:The mathematical expectation of the sum of several random variables is equal to the sum of their mathematical expectations.
Example. Let's calculate the mathematical expectation of a binomial random variable X – date of occurrence of the event A V n experiments.
Solution: Total number X occurrences of the event A in these trials is the sum of the number of occurrences of the event in individual trials. Let's introduce random variables X i– number of occurrences of the event in i th test, which are Bernoulli random variables with mathematical expectation, where . By the property of mathematical expectation we have
Thus, the mathematical expectation of a binomial distribution with parameters n and p is equal to the product np.
Example. Probability of hitting the target when firing a gun p = 0.6. Find the mathematical expectation of the total number of hits if 10 shots are fired.
Solution: The hit for each shot does not depend on the outcomes of other shots, therefore the events under consideration are independent and, consequently, the desired mathematical expectation
Each individual value is completely determined by its distribution function. Also, to solve practical problems, it is enough to know several numerical characteristics, thanks to which it becomes possible to present the main features of a random variable in a short form.
These quantities include primarily expected value And dispersion .
Expected value— the average value of a random variable in probability theory. Denoted as .
In the simplest way, the mathematical expectation of a random variable X(w), find how integralLebesgue in relation to the probability measure R original probability space
You can also find the mathematical expectation of a value as Lebesgue integral from X by probability distribution R X quantities X:
where is the set of all possible values X.
Mathematical expectation of functions from a random variable X found through distribution R X. For example, If X- a random variable with values in and f(x)- unambiguous Borel'sfunction X , That:
If F(x)- distribution function X, then the mathematical expectation is representable integralLebesgue - Stieltjes (or Riemann - Stieltjes):
in this case integrability X In terms of ( * ) corresponds to the finiteness of the integral
In specific cases, if X has a discrete distribution with probable values x k, k=1, 2, . , and probabilities, then
If X has an absolutely continuous distribution with probability density p(x), That
in this case, the existence of a mathematical expectation is equivalent to the absolute convergence of the corresponding series or integral.
Properties of the mathematical expectation of a random variable.
- The mathematical expectation of a constant value is equal to this value:
C- constant;
- M=C.M[X]
- The mathematical expectation of the sum of randomly taken values is equal to the sum of their mathematical expectations:
- The mathematical expectation of the product of independent randomly taken variables = the product of their mathematical expectations:
M=M[X]+M[Y]
If X And Y independent.
if the series converges:
Algorithm for calculating mathematical expectation.
Properties of discrete random variables: all their values can be renumbered by natural numbers; assign each value a non-zero probability.
1. Multiply the pairs one by one: x i on p i.
2. Add the product of each pair x i p i.
For example, For n = 4 :
Distribution function of a discrete random variable stepwise, it increases abruptly at those points whose probabilities have a positive sign.
Example: Find the mathematical expectation using the formula.
As is already known, the distribution law completely characterizes a random variable. However, often the distribution law is unknown and one has to limit oneself to less information. Sometimes it is even more profitable to use numbers that describe the random variable in total; such numbers are called numerical characteristics of a random variable. One of the important numerical characteristics is the mathematical expectation.
The mathematical expectation, as will be shown below, is approximately equal to the average value of the random variable. To solve many problems, it is enough to know the mathematical expectation. For example, if it is known that the mathematical expectation of the number of points scored by the first shooter is greater than that of the second, then the first shooter, on average, scores more points than the second, and, therefore, shoots better than the second. Although the mathematical expectation provides much less information about a random variable than the law of its distribution, knowledge of the mathematical expectation is sufficient for solving problems like the one above and many others.
§ 2. Mathematical expectation of a discrete random variable
Mathematical expectation A discrete random variable is the sum of the products of all its possible values and their probabilities.
Let the random variable X can only take values X 1 , X 2 , ..., X P , whose probabilities are respectively equal R 1 , R 2 , . . ., R P . Then the mathematical expectation M(X) random variable X is determined by equality
M(X) = X 1 R 1 + X 2 R 2 + … + x n p n .
If a discrete random variable X takes a countable set of possible values, then
M(X)=
Moreover, the mathematical expectation exists if the series on the right side of the equality converges absolutely.
Comment. From the definition it follows that the mathematical expectation of a discrete random variable is a non-random (constant) quantity. We recommend that you remember this statement, as it will be used many times later. It will be shown later that the mathematical expectation of a continuous random variable is also a constant value.
Example 1. Find the mathematical expectation of a random variable X, knowing the law of its distribution:
Solution. The required mathematical expectation is equal to the sum of the products of all possible values of the random variable and their probabilities:
M(X)= 3* 0, 1+ 5* 0, 6+ 2* 0, 3= 3, 9.
Example 2. Find the mathematical expectation of the number of occurrences of an event A in one trial, if the probability of the event A equal to R.
Solution. Random value X - number of occurrences of the event A in one test - can take only two values: X 1 = 1 (event A occurred) with probability R And X 2 = 0 (event A did not occur) with probability q= 1 -R. The required mathematical expectation
M(X)= 1* p+ 0* q= p
So, the mathematical expectation of the number of occurrences of an event in one trial is equal to the probability of this event. This result will be used below.
§ 3. Probabilistic meaning of mathematical expectation
Let it be produced P tests in which the random variable X accepted T 1 times value X 1 , T 2 times value X 2 ,...,m k times value x k , and T 1 + T 2 + …+t To = p. Then the sum of all values taken X, equal to
X 1 T 1 + X 2 T 2 + ... + X To T To .
Let's find the arithmetic mean 
all values accepted by a random variable, for which we divide the found sum by the total number of tests:
=
(X 1 T 1
+
X 2 T 2 +
... +
X To T To)/P,
=
X 1
(m 1 /
n)
+
X 2
(m 2 /
n)
+ ... +
X To
(T To /P).
(*)
Noticing that the attitude m 1 / n- relative frequency W 1 values X 1 , m 2 / n - relative frequency W 2 values X 2 etc., we write the relation (*) like this:
=X 1
W 1
+
x 2 W 2
+ ..
. + X To W k .
(**)
Let us assume that the number of tests is quite large. Then the relative frequency is approximately equal to the probability of the event occurring (this will be proven in Chapter IX, § 6):
W 1
p 1 ,
W 2
p 2 ,
…,
W k
p k .
Replacing the relative frequencies with the corresponding probabilities in relation (**), we obtain
x 1 p 1
+
X 2 R 2
+ … +
X To R To .
The right side of this approximate equality is M(X). So,
M(X).
The probabilistic meaning of the result obtained is as follows: mathematical expectation is approximately equal(the more accurate, the greater the number of tests) the arithmetic mean of the observed values of a random variable.
Remark 1. It is easy to understand that the mathematical expectation is greater than the smallest and less than the largest possible value. In other words, on the number line, possible values are located to the left and right of the mathematical expectation. In this sense, the mathematical expectation characterizes the location of the distribution and is therefore often called distribution center.
This term is borrowed from mechanics: if the masses R 1 , R 2 , ..., R P located at the abscissa points x 1 ,
X 2 ,
...,
X n, and 
then the abscissa of the center of gravity
x c =
.
Considering that
=
M
(X)
And 
we get M(X)= x With .
So, the mathematical expectation is the abscissa of the center of gravity of a system of material points, the abscissas of which are equal to the possible values of the random variable, and the masses are equal to their probabilities.
Remark 2. The origin of the term “mathematical expectation” is associated with the initial period of the emergence of probability theory (XVI - XVII centuries), when the scope of its application was limited to gambling. The player was interested in the average value of the expected win, or, in other words, the mathematical expectation of winning.
Mathematical expectation is the definition
Checkmate waiting is one of the most important concepts in mathematical statistics and probability theory, characterizing the distribution of values or probabilities random variable. Typically expressed as a weighted average of all possible parameters of a random variable. Widely used in technical analysis, the study of number series, and the study of continuous and time-consuming processes. It is important in assessing risks, predicting price indicators when trading on financial markets, and is used in developing strategies and methods of gaming tactics in gambling theories.
Checkmate waiting- This mean value of a random variable, distribution probabilities random variable is considered in probability theory.
Checkmate waiting is a measure of the average value of a random variable in probability theory. Checkmate the expectation of a random variable x denoted by M(x).
Mathematical expectation (Population mean) is
Checkmate waiting is
Checkmate waiting is in probability theory, a weighted average of all possible values that a random variable can take.
Checkmate waiting is the sum of the products of all possible values of a random variable and the probabilities of these values.
Mathematical expectation (Population mean) is
Checkmate waiting is the average benefit from a particular decision, provided that such a decision can be considered within the framework of the theory of large numbers and long distance.
Checkmate waiting is in gambling theory, the amount of winnings that a speculator can earn or lose, on average, on each bet. In the language of gambling speculators this is sometimes called "advantage" speculator" (if it is positive for the speculator) or "house edge" (if it is negative for the speculator).
Mathematical expectation (Population mean) is
Checkmate waiting is profit per win multiplied by the average profit, minus the loss, multiplied by the average loss.
Mathematical expectation of a random variable in mathematical theory
One of the important numerical characteristics of a random variable is the expected value. Let us introduce the concept of a system of random variables. Let's consider a set of random variables that are the results of the same random experiment. If is one of the possible values of the system, then the event corresponds to a certain probability that satisfies Kolmogorov’s axioms. A function defined for any possible values of random variables is called a joint distribution law. This function allows you to calculate the probabilities of any events from. In particular, joint law distributions of random variables and, which take values from the set and, are given by probabilities.
The term "mat. expectation" was introduced by Pierre Simon Marquis de Laplace (1795) and comes from the concept of "expected value of winnings", which first appeared in the 17th century in gambling theory in the works of Blaise Pascal and Christiaan Huygens. However, the first complete theoretical understanding and assessment of this concept was given by Pafnuty Lvovich Chebyshev (mid-19th century).
Law distributions of random numerical variables (distribution function and distribution series or probability density) completely describe the behavior of a random variable. But in a number of problems, it is enough to know some numerical characteristics of the quantity under study (for example, its average value and possible deviation from it) in order to answer the question posed. The main numerical characteristics of random variables are expectation, variance, mode and median.
The expectation of a discrete random variable is the sum of the products of its possible values and their corresponding probabilities. Sometimes swearing. the expectation is called a weighted average, since it is approximately equal to the arithmetic mean of the observed values of a random variable over a large number of experiments. From the definition of the expectation value it follows that its value is not less than the smallest possible value of the random variable and not greater than the largest. The expected value of a random variable is a non-random (constant) variable.
The mathematical expectation has a simple physical meaning: if you place a unit mass on a straight line, placing a certain mass at some points (for a discrete distribution), or “smearing” it with a certain density (for an absolutely continuous distribution), then the point corresponding to the mathematical expectation will be the coordinate "center of gravity" is straight.
The average value of a random variable is a certain number that is, as it were, its “representative” and replaces it in roughly approximate calculations. When we say: “the average lamp operating time is 100 hours” or “the average point of impact is shifted relative to the target by 2 m to the right,” we are indicating a certain numerical characteristic of a random variable that describes its location on the numerical axis, i.e. "position characteristics".
Of the characteristics of a position in probability theory, the most important role is played by the expected value of a random variable, which is sometimes called simply the average value of a random variable.
Consider the random variable X, having possible values x1, x2, …, xn with probabilities p1, p2, …, pn. We need to characterize with some number the position of the values of the random variable on the abscissa axis with taking into account that these values have different probabilities. For this purpose, it is natural to use the so-called “weighted average” of the values xi, and each value xi during averaging should be taken into account with a “weight” proportional to the probability of this value. Thus, we will calculate the average of the random variable X, which we denote M |X|:
This weighted average is called the expected value of a random variable. Thus, we introduced into consideration one of the most important concepts of probability theory - the concept of math. expectations. Mat. The expectation of a random variable is the sum of the products of all possible values of a random variable and the probabilities of these values.
Mat. waiting for a random variable X is connected by a peculiar dependence with the arithmetic mean of the observed values of the random variable over a large number of experiments. This dependence is of the same type as the dependence between frequency and probability, namely: with a large number of experiments, the arithmetic mean of the observed values of a random variable approaches (converges in probability) to its math. waiting. From the presence of a connection between frequency and probability, one can deduce as a consequence the presence of a similar connection between the arithmetic mean and the mathematical expectation. Indeed, consider the random variable X, characterized by a distribution series:
Let it be produced N independent experiments, in each of which the value X takes on a certain value. Let's assume that the value x1 appeared m1 times, value x2 appeared m2 times, general meaning xi appeared mi times. Let us calculate the arithmetic mean of the observed values of the value X, which, in contrast to the mathematical expectation M|X| we denote M*|X|:
With increasing number of experiments N frequencies pi will approach (converge in probability) the corresponding probabilities. Consequently, the arithmetic mean of the observed values of the random variable M|X| with an increase in the number of experiments it will approach (converge in probability) to its expected value. The connection formulated above between the arithmetic mean and math. expectation is the content of one of the forms of the law of large numbers.
We already know that all forms of the law of large numbers state the fact that some averages are stable over a large number of experiments. Here we are talking about the stability of the arithmetic mean from a series of observations of the same quantity. With a small number of experiments, the arithmetic mean of their results is random; with a sufficient increase in the number of experiments, it becomes “almost non-random” and, stabilizing, approaches a constant value - mat. waiting.
The stability of averages over a large number of experiments can be easily verified experimentally. For example, when weighing a body in a laboratory on precise scales, as a result of weighing we obtain a new value each time; To reduce observation error, we weigh the body several times and use the arithmetic mean of the obtained values. It is easy to see that with a further increase in the number of experiments (weighings), the arithmetic mean reacts to this increase less and less and, with a sufficiently large number of experiments, practically ceases to change.
It should be noted that the most important characteristic of the position of a random variable is mat. expectation - does not exist for all random variables. It is possible to create examples of such random variables for which mat. there is no expectation because the corresponding sum or integral diverges. However, such cases are not of significant interest for practice. Typically, the random variables we deal with have a limited range of possible values and, of course, have a mathematical expectation.
In addition to the most important of the characteristics of the position of a random variable - the expectation value - in practice, other characteristics of the position are sometimes used, in particular, the mode and median of the random variable.
The mode of a random variable is its most probable value. The term "most probable value" strictly speaking applies only to discontinuous quantities; for a continuous quantity, the mode is the value at which the probability density is maximum. The figures show the mode for discontinuous and continuous random variables, respectively.
If the distribution polygon (distribution curve) has more than one maximum, the distribution is called "multimodal".
Sometimes there are distributions that have a minimum in the middle rather than a maximum. Such distributions are called “anti-modal”.
In the general case, the mode and expected value of a random variable do not coincide. In the special case when the distribution is symmetric and modal (i.e. has a mode) and there is a mat. expectation, then it coincides with the mode and center of symmetry of the distribution.
Another position characteristic is often used - the so-called median of a random variable. This characteristic is usually used only for continuous random variables, although it can be formally defined for a discontinuous variable. Geometrically, the median is the abscissa of the point at which the area enclosed by the distribution curve is divided in half.
In the case of a symmetric modal distribution, the median coincides with the mat. expectation and fashion.
The expected value is the average value of a random variable - a numerical characteristic of the probability distribution of a random variable. In the most general way, checkmate the expectation of a random variable X(w) is defined as the Lebesgue integral with respect to the probability measure R in the original probability space:
Mat. the expectation can also be calculated as the Lebesgue integral of X by probability distribution px quantities X:
It is natural to define the concept of a random variable with infinite expectation. A typical example is the repatriation times in some random walks.
With the help of mat. expectations define many numerical and functional characteristics of the distribution (as the mathematical expectation of the corresponding functions from a random variable), for example, the generating function, characteristic function, moments of any order, in particular dispersion, covariance.
Mathematical expectation (Population mean) is
Mathematical expectation is a characteristic of the location of the values of a random variable (the average value of its distribution). In this capacity, the mathematical expectation serves as some “typical” distribution parameter and its role is similar to the role of the static moment - the coordinate of the center of gravity of the mass distribution - in mechanics. The expectation differs from other location characteristics with the help of which the distribution is described in general terms - medians, modes, mats - by the greater value that it and the corresponding scattering characteristic - dispersion - have in the limit theorems of probability theory. The meaning of the expectation mate is most fully revealed by the law of large numbers (Chebyshev's inequality) and the strengthened law of large numbers.
Mathematical expectation (Population mean) is
Expectation of a discrete random variable
Let there be some random variable that can take one of several numerical values (for example, the number of points when throwing a dice can be 1, 2, 3, 4, 5 or 6). Often in practice, for such a value, the question arises: what value does it take “on average” with a large number of tests? What will be our average income (or loss) from each of the risky transactions?
Let's say there is some kind of lottery. We want to understand whether it is profitable or not to participate in it (or even participate repeatedly, regularly). Let’s say that every fourth ticket is a winner, the prize will be 300 rubles, and any ticket will be 100 rubles. With an infinitely large number of participations, this is what happens. In three quarters of cases we will lose, every three losses will cost 300 rubles. In every fourth case we will win 200 rubles. (prize minus cost), that is, for four participations we lose on average 100 rubles, for one - on average 25 rubles. In total, the average rate of our ruin will be 25 rubles per ticket.
We throw the dice. If it is not cheating (without shifting the center of gravity, etc.), then how many points will we have on average at a time? Since each option is equally likely, we simply take the arithmetic mean and get 3.5. Since this is AVERAGE, there is no need to be indignant that no specific roll will give 3.5 points - well, this cube does not have a face with such a number!
Now let's summarize our examples:
Let's look at the picture just given. On the left is a table of the distribution of a random variable. The value X can take one of n possible values (shown in the top line). There cannot be any other meanings. Under each possible value, its probability is written below. On the right is the formula, where M(X) is called mat. waiting. The meaning of this value is that with a large number of tests (with a large sample), the average value will tend to this same expectation.
Let's return again to the same playing cube. Mat. the expected number of points when throwing is 3.5 (calculate it yourself using the formula if you don’t believe me). Let's say you threw it a couple of times. The results were 4 and 6. The average was 5, which is far from 3.5. They threw it one more time, they got 3, that is, on average (4 + 6 + 3)/3 = 4.3333... Somehow far from the mat. expectations. Now do a crazy experiment - roll the cube 1000 times! And even if the average is not exactly 3.5, it will be close to that.
Let's calculate the mat. waiting for the lottery described above. The plate will look like this:
Then the checkmate expectation will be as we established above:
Another thing is that doing it “on the fingers”, without a formula, would be difficult if there were more options. Well, let's say there would be 75% losing tickets, 20% winning tickets and 5% especially winning ones.
Now some properties mat expectations.
Mat. the expectation is linear. It's easy to prove:
The constant multiplier can be taken out beyond the checkmate sign. expectations, that is:
This is a special case of the linearity property of the expectation mate.
Another consequence of the linearity of mat. expectations:
that is, mat. the expectation of the sum of random variables is equal to the sum of the mathematical expectations of random variables.
Let X, Y be independent random variables, Then:
This is also easy to prove) Work XY itself is a random variable, and if the initial values could take n And m values accordingly, then XY can take nm values. each value is calculated based on the fact that the probabilities of independent events are multiplied. As a result, we get this:
Expectation of a continuous random variable
Continuous random variables have such a characteristic as distribution density (probability density). It essentially characterizes the situation that a random variable takes some values from the set of real numbers more often, and some less often. For example, consider this graph:
Here X- actual random variable, f(x)- distribution density. Judging by this graph, during experiments the value X will often be a number close to zero. The chances are exceeded 3 or be smaller -3 rather purely theoretical.
If the distribution density is known, then the expected value is found as follows:
Let, for example, there be a uniform distribution:
Let's find a checkmate. expectation:
This is quite consistent with intuitive understanding. Let's say, if we receive many random real numbers with a uniform distribution, each of the segment |0; 1| , then the arithmetic mean should be about 0.5.
The properties of mathematical expectations - linearity, etc., applicable for discrete random variables, are also applicable here.
Relationship between mathematical expectation and other statistical indicators
IN statistical analysis, along with mathematical expectation, there is a system of interdependent indicators that reflect the homogeneity of phenomena and stability processes. Variation indicators often have no independent meaning and are used for further data analysis. The exception is the coefficient of variation, which characterizes the homogeneity data what is valuable statistical characteristic.
Degree of variability or stability processes in statistical science can be measured using several indicators.
The most important indicator characterizing variability random variable is Dispersion, which is most closely and directly related to mat. waiting. This parameter is actively used in other types of statistical analysis (hypothesis testing, analysis of cause-and-effect relationships, etc.). Like the average linear deviation, dispersion also reflects the measure of spread data around the average value.
It is useful to translate the language of signs into the language of words. It turns out that the dispersion is the average square of the deviations. That is, the average value is first calculated, then the difference between each original and average value is taken, squared, added, and then divided by the number of values in the population. Difference between an individual value and the average reflects the measure of deviation. It is squared so that all deviations become exclusively positive numbers and to avoid mutual destruction of positive and negative deviations when summing them up. Then, given the squared deviations, we simply calculate the arithmetic mean. Mean - square - deviations. Deviations are squared and the average is calculated. The answer to the magic word “dispersion” lies in just three words.
However, in its pure form, such as the arithmetic mean, or dispersion, is not used. It is rather an auxiliary and intermediate indicator that is used for other types of statistical analysis. It doesn't even have a normal unit of measurement. Judging by the formula, this is the square of the unit of measurement of the original data.
Mathematical expectation (Population mean) is
Let us measure a random variable N times, for example, we measure the wind speed ten times and want to find the average value. How is the average value related to the distribution function?
Or we will roll the dice a large number of times. The number of points that will appear on the dice with each throw is a random variable and can take any natural value from 1 to 6. The arithmetic mean of the dropped points calculated for all dice throws is also a random variable, but for large N it tends to a very specific number - checkmate. waiting Mx. In this case Mx = 3.5.
How did you get this value? Let in N tests n1 1 point is rolled once n2 once - 2 points and so on. Then the number of outcomes in which one point fell:
Similarly for outcomes when 2, 3, 4, 5 and 6 points are rolled.
Let us now assume that we know the distributions of the random variable x, that is, we know that the random variable x can take values x1, x2,..., xk with probabilities p1, p2,..., pk.
Math expectation Mx of random variable x is equal to:
Math expectation is not always a reasonable estimate of some random variable. So, to estimate the average salary, it is more reasonable to use the concept of median, that is, such a value that the number of people earning less than the median salary and large, coincide.
The probability p1 that the random variable x will be less than x1/2, and the probability p2 that the random variable x will be greater than x1/2, are the same and equal to 1/2. The median is not determined uniquely for all distributions.
Standard or Standard Deviation in statistics, the degree of deviation of observational data or sets from the AVERAGE value is called. Denoted by the letters s or s. A small standard deviation indicates that the data clusters around the mean, while a large standard deviation indicates that the initial data are located far from it. The standard deviation is equal to the square root of a quantity called variance. It is the average of the sum of the squared differences of the initial data that deviate from the average value. The standard deviation of a random variable is the square root of the variance:
Example. Under test conditions when shooting at a target, calculate the dispersion and standard deviation of the random variable:
Variation- fluctuation, changeability of the value of a characteristic among units of the population. Individual numerical values of a characteristic found in the population being studied are called variant values. The insufficiency of the average value to fully characterize the population forces us to supplement the average values with indicators that allow us to assess the typicality of these averages by measuring the variability (variation) of the characteristic being studied. The coefficient of variation is calculated using the formula:
Range of variation(R) represents the difference between the maximum and minimum values of the attribute in the population being studied. This indicator gives the most general idea of the variability of the characteristic being studied, as it shows difference only between the extreme values of the options. Dependence on the extreme values of a characteristic gives the scope of variation an unstable, random character.
Average linear deviation represents the arithmetic mean of the absolute (modulo) deviations of all values of the analyzed population from their average value:
Expectation in gambling theory
Checkmate waiting is the average amount of money a gambling speculator can win or lose on a given bet. This is a very important concept for a speculator because it is fundamental to the assessment of most gambling situations. Checkmate is also an optimal tool for analyzing basic card layouts and gaming situations.
Let's say you're playing a coin game with a friend, betting equally $1 each time, no matter what comes up. Tails means you win, heads you lose. The odds are one to one that it will come up heads, so you bet $1 to $1. Thus, your checkmate expectation is equal to zero, because From a mathematical point of view, you cannot know whether you will lead or lose after two throws or after 200.
Your hourly gain is zero. Hourly winnings are the amount of money you expect to win in an hour. You can toss a coin 500 times in an hour, but you won't win or lose because... your chances are neither positive nor negative. From the point of view of a serious speculator, this betting system is not bad. But this is simply a waste of time.
But let's say someone wants to bet $2 against your $1 on the same game. Then you immediately have a positive expectation of 50 cents from each bet. Why 50 cents? On average, you win one bet and lose the second. Bet first and you will lose $1, bet second and you will win $2. You bet $1 twice and are ahead by $1. So each of your one dollar bets gave you 50 cents.
If a coin appears 500 times in one hour, your hourly winnings will already be $250, because... on average you lost one dollar 250 times and won two dollar 250 times. $500 minus $250 equals $250, which is the total winnings. Please note that the expected value, which is the average amount you win per bet, is 50 cents. You won $250 by betting a dollar 500 times, which equals 50 cents per bet.
Mathematical expectation (Population mean) is
Mat. waiting has nothing to do with short-term results. Your opponent, who decided to bet $2 against you, could beat you on the first ten rolls in a row, but you, having a 2 to 1 betting advantage, all other things being equal, will earn 50 cents on every $1 bet in any circumstances. It makes no difference whether you win or lose one bet or several bets, as long as you have enough cash to comfortably cover the costs. If you continue to bet in the same way, then over a long period of time your winnings will approach the sum of the expectations in individual throws.
Every time you make a best bet (a bet that may turn out to be profitable in the long run), when the odds are in your favor, you are bound to win something on it, no matter whether you lose it or not in the given hand. Conversely, if you make an underdog bet (a bet that is unprofitable in the long run) when the odds are against you, you lose something regardless of whether you win or lose the hand.
Mathematical expectation (Population mean) is
You place a bet with the best outcome if your expectation is positive, and it is positive if the odds are on your side. When you place a bet with the worst outcome, you have a negative expectation, which happens when the odds are against you. Serious speculators only bet on the best outcome; if the worst happens, they fold. What does the odds mean in your favor? You may end up winning more than the real odds bring. The real odds of landing heads are 1 to 1, but you get 2 to 1 due to the odds ratio. In this case, the odds are in your favor. You definitely get the best outcome with a positive expectation of 50 cents per bet.
Here is a more complex example of mat. expectations. A friend writes down numbers from one to five and bets $5 against your $1 that you won't guess the number. Should you agree to such a bet? What is the expectation here?
On average you will be wrong four times. Based on this, the odds against you guessing the number are 4 to 1. The odds against you losing a dollar on one attempt. However, you win 5 to 1, with the possibility of losing 4 to 1. So the odds are in your favor, you can take the bet and hope for the best outcome. If you make this bet five times, on average you will lose $1 four times and win $5 once. Based on this, for all five attempts you will earn $1 with a positive mathematical expectation of 20 cents per bet.
A speculator who expects to win more than he bets, as in the example above, is taking chances. On the contrary, he ruins his chances when he expects to win less than he bets. A speculator placing a bet can have either a positive or negative expectation, which depends on whether he wins or ruins the odds.
If you bet $50 to win $10 with a 4 to 1 chance of winning, you will get a negative expectation of $2 because On average, you will win $10 four times and lose $50 once, which shows that the loss per bet will be $10. But if you bet $30 to win $10, with the same odds of winning 4 to 1, then in this case you have a positive expectation of $2, because you again win four times $10 and lose once $30, which is profit at $10. These examples show that the first bet is bad, and the second is good.
Mat. anticipation is the center of any game situation. When a bookmaker encourages football fans to bet $11 to win $10, he has a positive expectation of 50 cents on every $10. If the casino pays even money from the pass line in craps, then the casino's positive expectation will be approximately $1.40 for every $100, because This game is structured so that anyone who bets on this line loses 50.7% on average and wins 49.3% of the total time. Undoubtedly, it is this seemingly minimal positive expectation that brings colossal profits to casino owners around the world. As Vegas World casino owner Bob Stupak noted, “one thousandth percent negative probability over a long enough distance will ruin the richest man in the world.”
Expectation when playing Poker
The game of Poker is the most illustrative and illustrative example from the point of view of using the theory and properties of expectation mate.
Mat. Expected Value in Poker is the average benefit from a particular decision, provided that such a decision can be considered within the framework of the theory of large numbers and long distance. A successful poker game is to always accept moves with positive expected value.
Mathematical expectation (Population mean) is
Mathematical meaning of math. The expectation when playing poker is that we often encounter random variables when making decisions (we don’t know exactly what cards the opponent has in his hands, what cards will come on subsequent rounds trade). We must consider each of the solutions from the point of view of large number theory, which states that with a sufficiently large sample, the average value of a random variable will tend to its expected value.
Among the particular formulas for calculating mate expectations, the following is most applicable in poker:
When playing poker checkmate. expectation can be calculated for both bets and calls. In the first case, fold equity should be taken into account, in the second, the bank’s own odds. When assessing mat. expectations of a particular move, it should be remembered that a fold always has a zero expectation. Thus, discarding cards will always be a more profitable decision than any negative move.
Mathematical expectation (Population mean) is
Expectation tells you what you can expect (or loss) for every risk you take. Casinos make money money, since checkmate is an expectation from all games that are practiced in them, in favor of the casino. With a long enough series of games, you can expect the client to lose his money, since the “odds” are in favor of the casino. However, professional casino speculators limit their games to short periods of time, thereby increasing the odds in their favor. The same goes for investing. If your expectation is positive, you can make more money by making many trades in a short time period time. Expectation is your percentage of profit per win multiplied by your average profit, minus your probability of loss multiplied by your average loss.
Poker can also be viewed from the standpoint of checkmate expectations. You may assume that a certain move is profitable, but in some cases it may not be the best because another move is more profitable. Let's say you hit a full house in five-card draw poker. Your opponent makes a bet. You know that if you raise the bet, he will respond. Therefore, raising seems to be the best tactic. But if you do raise the bet, the remaining two speculators will definitely fold. But if you call, you have full confidence that the other two speculators after you will do the same. When you raise your bet you get one unit, and when you just call you get two. Thus, calling gives you a higher positive expected value and will be the best tactic.
Mat. expectation can also give an idea of which poker tactics are less profitable and which ones are more profitable. For example, if you play a certain hand and you think your loss will average 75 cents including ante, then you should play that hand because this is better than folding when the ante is $1.
Another important reason for understanding the essence of mate. expectation is that it gives you a sense of peace whether you win the bet or not: if you made a good bet or folded at the right time, you will know that you have earned or saved a certain amount of money that a weaker speculator could not save. It's much harder to fold if you're upset because your opponent drew a stronger hand. With all this, what you saved by not playing, instead of betting, is added to your winnings per night or per month.
Just remember that if you changed your hands, your opponent would have called you, and as you will see in the Fundamental Theorem of Poker article, this is just one of your advantages. You should be happy when this happens. You may even learn to enjoy losing a hand because you know that other speculators in your position would have lost much more.
As mentioned in the coin game example at the beginning, the hourly profit ratio is interconnected with the expected maturation, and this concept is especially important for professional speculators. When you go to play poker, you should mentally estimate how much you can win in an hour of play. In most cases you will need to rely on your intuition and experience, but you can also use some math. For example, you are playing draw lowball and you see three players bet $10 and then trade two cards, which is a very bad tactic, you can figure out that every time they bet $10, they lose about $2. Each of them does this eight times per hour, which means that all three of them lose approximately $48 per hour. You are one of the remaining four speculators, who are approximately equal, so these four speculators (and you among them) must split $48, each making a profit of $12 per hour. Your hourly odds in this case are simply equal to your share of the amount of money lost by three bad speculators in an hour.
Mathematical expectation (Population mean) is
Over a long period of time, the total winnings of a speculator is the sum of his mathematical expectations in individual hands. The more hands you play with positive expectation, the more you win, and conversely, the more hands you play with negative expectation, the more you lose. As a result, you should choose a game that can maximize your positive expectation or negate your negative expectation so that you can maximize your hourly winnings.
Positive mathematical expectation in gaming strategy
If you know how to count cards, you can have an advantage over the casino if they don't notice and throw you out. Casinos love drunken speculators and can't stand card counting. An advantage will allow you to win more times than you lose over time. Good money management when using expectation mate calculations can help you extract more profit from your advantage and reduce your losses. Without an advantage, you're better off giving the money to charity. In the game on the stock exchange, an advantage is given by the game system that creates greater profits than losses, the difference prices and commissions. None capital Management will not save a bad gaming system.
A positive expectation is defined as a value greater than zero. The larger this number, the stronger the statistical expectation. If the value is less than zero, then checkmate. the expectation will also be negative. The larger the module of the negative value, the worse the situation. If the result is zero, then the wait is break-even. You can only win when you have a positive mathematical expectation and a reasonable playing system. Playing by intuition leads to disaster.
Mathematical expectation and
Checkmate expectation is a fairly widely demanded and popular statistical indicator when carrying out exchange trading on financial markets. First of all, this parameter is used to analyze the success of trade. It is not difficult to guess that the higher this value, the more reasons to consider the trade being studied successful. Of course, analysis work the trader cannot be made only using this parameter. However, the calculated value in combination with other methods of assessing quality work, can significantly improve the accuracy of the analysis.
The expectation checkmate is often calculated in trading account monitoring services, which allows you to quickly evaluate the work performed on the deposit. The exceptions include strategies that use “sitting out” unprofitable trades. Trader luck may accompany him for some time, and therefore there may be no losses in his work at all. In this case, it will not be possible to be guided only by the mathematical expectation, because the risks used in the work will not be taken into account.
In trading on market checkmate is most often used when predicting the profitability of any trading strategy or when forecasting income trader based on statistical data from his previous bidding.
Mathematical expectation (Population mean) is
In relation to money management, it is very important to understand that there is no pattern when making trades with negative expectation management money, which can definitely bring high profits. If you keep playing stock exchange under these conditions, then regardless of the method management money, you will lose your entire account, no matter how large it was at the beginning.
This axiom is true not only for games or trades with negative expectation, it is also true for games with equal chances. Therefore, the only time you have a chance to profit in the long term is if you take trades with positive expected value.
The difference between negative expectation and positive expectation is the difference between life and death. It doesn't matter how positive or how negative the expectation is; All that matters is whether it is positive or negative. Therefore, before considering management issues capital you have to find a game with positive anticipation.
If you don't have such a game, then all the money management in the world will not save you. On the other hand, if you have a positive expectation, you can, through proper money management, turn it into an exponential growth function. It doesn't matter how small the positive expectation is! In other words, it doesn't matter how profitable a trading system is based on a single contract. If you have a system that wins $10 per contract per trade (after commissions and slippage), you can use management techniques capital in a way that makes it more profitable than a system that shows an average profit of $1,000 per trade (after commissions and slippage).
What matters is not how profitable the system was, but how certain the system can be said to show at least minimal profit in the future. Therefore, the most important preparation that can be made is to ensure that the system will show a positive expected value in the future.
In order to have a positive expected value in the future, it is very important not to limit the degrees of freedom of your system. This is achieved not only by eliminating or reducing the number of parameters to be optimized, but also by reducing as many system rules as possible. Every parameter you add, every rule you make, every tiny change you make to the system reduces the number of degrees of freedom. Ideally, you need to build a fairly primitive and simple system that will consistently generate small profits in almost any market. Again, it is important for you to understand that it does not matter how profitable the system is, as long as it is profitable. The money you earn in trading will be earned through effective money management.
Mathematical expectation (Population mean) is
A trading system is simply a tool that gives you a positive expected value so that you can use money management. Systems that work (show at least minimal profits) in only one or a few markets, or have different rules or parameters for different markets, will most likely not work in real time for long. The problem with most technically oriented traders is that they spend too much time and effort optimizing the various rules and parameter values of the trading system. This gives completely opposite results. Instead of wasting energy and computer time on increasing the profits of the trading system, direct your energy to increasing the level of reliability of obtaining a minimum profit.
Knowing that capital Management is just a numbers game that requires the use of positive expectations, a trader can stop searching for the "holy grail" of stock trading. Instead, he can start testing his trading method, find out how logical this method is, and whether it gives positive expectations. Proper money management methods, applied to any, even very mediocre trading methods, will do the rest of the work themselves.
For any trader to succeed in his work, he needs to solve three most important tasks:. To ensure that the number of successful transactions exceeds the inevitable mistakes and miscalculations; Set up your trading system so that you have the opportunity to earn money as often as possible; Achieve stable positive results from your operations.
And here, for us working traders, mate can be a good help. expectation. This term is one of the key ones in probability theory. With its help, you can give an average estimate of some random value. The expectation of a random variable is similar to the center of gravity, if you imagine all possible probabilities as points with different masses.
In relation to a trading strategy, the expectation of profit (or loss) is most often used to evaluate its effectiveness. This parameter is defined as the sum of the products of the given levels of profit and loss and the probability of their occurrence. For example, the developed trading strategy assumes that 37% of all transactions will bring profit, and the remaining part - 63% - will be unprofitable. At the same time, the average income from a successful trade will be $7, and the average loss will be $1.4. Let's calculate the math. expectation of trading using this system:
What does this number mean? It says that, following the rules of this system, on average we will receive $1,708 from each closed transaction. Since the resulting efficiency rating is greater than zero, such a system can be used for real work. If, as a result of calculating the checkmate, the expectation turns out to be negative, then this already indicates an average loss and this will lead to ruin.
The amount of profit per transaction can also be expressed as a relative value in the form of %. For example:
The percentage of income per 1 transaction is 5%;
The percentage of successful trading operations is 62%;
Loss percentage per 1 trade - 3%;
The percentage of unsuccessful transactions is 38%;
In this case, checkmate. the expectation will be:
That is, the average trade will bring 1.96%.
It is possible to develop a system that, despite the predominance of unprofitable trades, will produce a positive result, since its MO>0.
However, waiting alone is not enough. It is difficult to make money if the system gives very few trading signals. In this case, it will be comparable to bank interest. Let each operation produce on average only 0.5 dollars, but what if the system involves 1000 operations per year? This will be a very significant amount in a relatively short time. It logically follows from this that another distinctive feature of a good trading system can be considered a short period of holding positions.
Sources and links
dic.academic.ru - academic online dictionary
mathematics.ru - educational website in mathematics
nsu.ru - educational website of Novosibirsk State University
webmath.ru is an educational portal for students, applicants and schoolchildren.
exponenta.ru educational mathematical website
ru.tradimo.com - free online trading school
crypto.hut2.ru - multidisciplinary information resource
poker-wiki.ru - free encyclopedia of poker
sernam.ru - Scientific library of selected natural science publications
reshim.su - website WE WILL SOLVE test coursework problems
unfx.ru - Forex on UNFX: training, trading signals, trust management
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