Characteristics of DSW and their properties. Mathematical expectation, variance, standard deviation

The distribution law fully characterizes the random variable. However, when it is impossible to find the distribution law, or this is not required, one can limit oneself to finding values, called numerical characteristics of a random variable. These quantities determine some average value around which the values ​​of a random variable are grouped, and the degree of their dispersion around this average value.

mathematical expectation A discrete random variable is the sum of the products of all possible values ​​of a random variable and their probabilities.

The mathematical expectation exists if the series on the right side of the equality converges absolutely.

In terms of probability, we can say that expected value approximately equal to the arithmetic mean of the observed values ​​of the random variable.

Example. The law of distribution of a discrete random variable is known. Find the mathematical expectation.

X
p	0.2	0.3	0.1	0.4

Solution:

9.2 Expectation Properties

1. Mathematical expectation constant value equal to the most constant.

2. A constant factor can be taken out of the expectation sign.

3. The mathematical expectation of the product of two independent random variables is equal to the product of their mathematical expectations.

This property is valid for an arbitrary number of random variables.

4. The mathematical expectation of the sum of two random variables is equal to the sum of the mathematical expectations of the terms.

This property is also true for an arbitrary number of random variables.

Let n independent trials be performed, the probability of occurrence of event A in which is equal to p.

Theorem. The mathematical expectation M(X) of the number of occurrences of event A in n independent trials is equal to the product of the number of trials and the probability of occurrence of the event in each trial.

Example. Find the mathematical expectation of a random variable Z if the mathematical expectations of X and Y are known: M(X)=3, M(Y)=2, Z=2X+3Y.

Solution:

9.3 Dispersion of a discrete random variable

However, the mathematical expectation cannot fully characterize a random process. In addition to the mathematical expectation, it is necessary to introduce a value that characterizes the deviation of the values ​​of the random variable from the mathematical expectation.

This deviation is equal to the difference between the random variable and its mathematical expectation. In this case, the mathematical expectation of the deviation is zero. This is explained by the fact that some possible deviations are positive, others are negative, and as a result of their mutual cancellation, zero is obtained.

Dispersion (scattering) Discrete random variable is called the mathematical expectation of the squared deviation of the random variable from its mathematical expectation.

In practice, this method of calculating the variance is inconvenient, because leads to cumbersome calculations for a large number of values ​​of a random variable.

Therefore, another method is used.

Theorem. The variance is equal to the difference between the mathematical expectation of the square of the random variable X and the square of its mathematical expectation.

Proof. Taking into account the fact that the mathematical expectation M (X) and the square of the mathematical expectation M 2 (X) are constant values, we can write:

Example. Find the variance of a discrete random variable given by the distribution law.

X
X 2
R	0.2	0.3	0.1	0.4

Solution: .

9.4 Dispersion properties

1. The dispersion of a constant value is zero. .

2. A constant factor can be taken out of the dispersion sign by squaring it. .

3. The variance of the sum of two independent random variables is equal to the sum of the variances of these variables. .

4. The variance of the difference of two independent random variables is equal to the sum of the variances of these variables. .

Theorem. The variance of the number of occurrences of event A in n independent trials, in each of which the probability p of the occurrence of the event is constant, is equal to the product of the number of trials and the probabilities of occurrence and non-occurrence of the event in each trial.

9.5 Standard deviation of a discrete random variable

Standard deviation random variable X is called the square root of the variance.

Theorem. The root mean square deviation of the sum of a finite number of mutually independent random variables is square root from the sum of the squares of the standard deviations of these quantities.

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Definition

Let a probability space be given (Ω , A , P) (\displaystyle (\Omega ,(\mathfrak (A)),\mathbb (P))) and the random value defined on it X (\displaystyle X). That is, by definition, X: Ω → R (\displaystyle X\colon \Omega \to \mathbb (R) ) is a measurable function. If there exists a Lebesgue integral of X (\displaystyle X) by space Ω (\displaystyle \Omega ), then it is called the mathematical expectation, or the average (expected) value and is denoted M [ X ] (\displaystyle M[X]) or E [ X ] (\displaystyle \mathbb (E) [X]).

M [ X ] = ∫ Ω X (ω) P (d ω) . (\displaystyle M[X]=\int \limits _(\Omega )\!X(\omega)\,\mathbb (P) (d\omega).)

Basic formulas for mathematical expectation

M [ X ] = ∫ − ∞ ∞ x d F X (x) ; x ∈ R (\displaystyle M[X]=\int \limits _(-\infty )^(\infty )\!x\,dF_(X)(x);x\in \mathbb (R) ).

Mathematical expectation of a discrete distribution

P (X = x i) = p i , ∑ i = 1 ∞ p i = 1 (\displaystyle \mathbb (P) (X=x_(i))=p_(i),\;\sum \limits _(i=1 )^(\infty )p_(i)=1),

then it follows directly from the definition of the Lebesgue integral that

M [ X ] = ∑ i = 1 ∞ x i p i (\displaystyle M[X]=\sum \limits _(i=1)^(\infty )x_(i)\,p_(i)).

Mathematical expectation of an integer value

P (X = j) = p j , j = 0 , 1 , . . . ; ∑ j = 0 ∞ p j = 1 (\displaystyle \mathbb (P) (X=j)=p_(j),\;j=0,1,...;\quad \sum \limits _(j=0 )^(\infty )p_(j)=1)

then its mathematical expectation can be expressed in terms of the generating function of the sequence ( p i ) (\displaystyle \(p_(i)\))

P (s) = ∑ k = 0 ∞ p k s k (\displaystyle P(s)=\sum _(k=0)^(\infty )\;p_(k)s^(k))

as the value of the first derivative at unity: M [ X ] = P ′ (1) (\displaystyle M[X]=P"(1)). If the mathematical expectation X (\displaystyle X) infinite, then lim s → 1 P ′ (s) = ∞ (\displaystyle \lim _(s\to 1)P"(s)=\infty ) and we will write P ′ (1) = M [ X ] = ∞ (\displaystyle P"(1)=M[X]=\infty )

Now let's take the generating function Q (s) (\displaystyle Q(s)) sequences of "tails" of the distribution ( q k ) (\displaystyle \(q_(k)\))

q k = P (X > k) = ∑ j = k + 1 ∞ p j ; Q (s) = ∑ k = 0 ∞ q k s k . (\displaystyle q_(k)=\mathbb (P) (X>k)=\sum _(j=k+1)^(\infty )(p_(j));\quad Q(s)=\sum _(k=0)^(\infty )\;q_(k)s^(k).)

This generating function is related to the previously defined function P (s) (\displaystyle P(s)) property: Q (s) = 1 − P (s) 1 − s (\displaystyle Q(s)=(\frac (1-P(s))(1-s))) at | s |< 1 {\displaystyle |s|<1} . From this, according to the mean-value theorem, it follows that the mathematical expectation is simply equal to the value of this function at unity:

M [ X ] = P ′ (1) = Q (1) (\displaystyle M[X]=P"(1)=Q(1))

The mathematical expectation of an absolutely continuous distribution

M [ X ] = ∫ − ∞ ∞ x f X (x) d x (\displaystyle M[X]=\int \limits _(-\infty )^(\infty )\!xf_(X)(x)\,dx ).

Mathematical expectation of a random vector

Let X = (X 1 , … , X n) ⊤ : Ω → R n (\displaystyle X=(X_(1),\dots ,X_(n))^(\top )\colon \Omega \to \mathbb ( R) ^(n)) is a random vector. Then by definition

M [ X ] = (M [ X 1 ] , … , M [ X n ]) ⊤ (\displaystyle M[X]=(M,\dots ,M)^(\top )),

that is, the mathematical expectation of a vector is determined component by component.

Mathematical expectation of the transformation of a random variable

Let g: R → R (\displaystyle g\colon \mathbb (R) \to \mathbb (R) ) is a Borel function such that the random variable Y = g(X) (\displaystyle Y=g(X)) has a finite mathematical expectation. Then the formula is valid for it:

M [ g (X) ] = ∑ i = 1 ∞ g (x i) p i (\displaystyle M\left=\sum \limits _(i=1)^(\infty )g(x_(i))p_(i )),

if X (\displaystyle X) has a discrete distribution;

M [ g (X) ] = ∫ − ∞ ∞ g (x) f X (x) d x (\displaystyle M\left=\int \limits _(-\infty )^(\infty )\!g(x) f_(X)(x)\,dx),

if X (\displaystyle X) has an absolutely continuous distribution.

If the distribution P X (\displaystyle \mathbb (P) ^(X)) random variable X (\displaystyle X) general form, then

M [ g (X) ] = ∫ − ∞ ∞ g (x) P X (d x) (\displaystyle M\left=\int \limits _(-\infty )^(\infty )\!g(x)\, \mathbb (P) ^(X)(dx)).

In the special case when g (X) = X k (\displaystyle g(X)=X^(k)), Expected value M [ g (X) ] = M [ X k ] (\displaystyle M\left=M) called k (\displaystyle k)-th moment of the random variable.

The simplest properties of mathematical expectation

• The mathematical expectation of a number is the number itself.

M [ a ] ​​= a (\displaystyle M[a]=a) a ∈ R (\displaystyle a\in \mathbb (R) )- constant;

• The mathematical expectation is linear, that is

M [ a X + b Y ] = a M [ X ] + b M [ Y ] (\displaystyle M=aM[X]+bM[Y]), where X , Y (\displaystyle X,Y) are random variables with a finite mathematical expectation, and a , b ∈ R (\displaystyle a,b\in \mathbb (R) )- arbitrary constants; 0 ⩽ M [ X ] ⩽ M [ Y ] (\displaystyle 0\leqslant M[X]\leqslant M[Y]); M [ X ] = M [ Y ] (\displaystyle M[X]=M[Y]). M [ X Y ] = M [ X ] M [ Y ] (\displaystyle M=M[X]M[Y]).

1. The mathematical expectation of a constant value is equal to the constant itself M(S)=S .
2. A constant factor can be taken out of the expectation sign: M(CX)=CM(X)
3. The mathematical expectation of the product of two independent random variables is equal to the product of their mathematical expectations: M(XY)=M(X) M(Y).
4. The mathematical expectation of the sum of two random variables is equal to the sum of the mathematical expectations of the terms: M(X+Y)=M(X)+M(Y).

Theorem. The mathematical expectation M(x) of the number of occurrences of events A in n independent trials is equal to the product of these trials by the probability of occurrence of events in each trial: M(x) = np.

Let X is a random variable and M(X) is its mathematical expectation. Consider as a new random variable the difference X - M(X).

Deviation is the difference between a random variable and its mathematical expectation.

The deviation has the following distribution law:

Solution: Find the mathematical expectation:
2 =(1-2.3) 2 =1.69
2 =(2-2.3) 2 =0.09
2 =(5-2.3) 2 =7.29

Let's write the distribution law of the squared deviation:

Solution: Find the expectation M(x): M(x)=2 0.1+3 0.6+5 0.3=3.5

Let's write the distribution law of the random variable X 2

x2
P	0.1	0.6	0.3

Let's find the mathematical expectation M(x2):M(x2) = 4 0.1+9 0.6+25 0.3=13.5

The desired dispersion D (x) \u003d M (x 2) - 2 \u003d 13.3- (3.5) 2 \u003d 1.05

Dispersion properties:

1. Dispersion of a constant value FROM equals zero: D(C)=0
2. A constant factor can be taken out of the dispersion sign by squaring it. D(Cx)=C 2 D(x)
3. The variance of the sum of independent random variables is equal to the sum of the variances of these variables. D(X 1 +X 2 +...+X n)=D(X 1)+D(X 2)+...+D(X n)
4. The variance of the binomial distribution is equal to the product of the number of trials and the probability of occurrence and non-occurrence of an event in one trial D(X)=npq

To estimate the dispersion of possible values ​​of a random variable around its mean value, in addition to the variance, some other characteristics also serve. Among them is the standard deviation.

The standard deviation of a random variable X called the square root of the variance:

σ(X) = √D(X) (4)

Example. The random variable X is given by the distribution law

X
P	0.1	0.4	0.5

Find the standard deviation σ(x)

Solution: Find the mathematical expectation X: M(x)=2 0.1+3 0.4+10 0.5=6.4
Let's find the mathematical expectation of X 2: M(x 2)=2 2 0.1+3 2 0.4+10 2 0.5=54
Find the dispersion: D(x)=M(x 2)=M(x 2)- 2 =54-6.4 2 =13.04
Desired standard deviation σ(X)=√D(X)=√13.04≈3.61

Theorem. The standard deviation of the sum of a finite number of mutually independent random variables is equal to the square root of the sum of the squared standard deviations of these variables:

Example. There are 3 books on mathematics and 3 on physics on a shelf of 6 books. Three books are chosen at random. Find the law of distribution of the number of books in mathematics among the selected books. Find the mathematical expectation and variance of this random variable.

The mathematical expectation is, the definition

Mat waiting is one of the most important concepts in mathematical statistics and probability theory, characterizing the distribution of values ​​or probabilities random variable. Usually expressed as a weighted average of all possible parameters of a random variable. It is widely used in technical analysis, the study of number series, the study of continuous and long-term processes. It is important in assessing risks, predicting price indicators when trading in financial markets, and is used in the development of strategies and methods of game tactics in gambling theory.

Checkmate waiting- this is mean value of a random variable, distribution probabilities random variable is considered in probability theory.

Mat waiting is measure of the mean value of a random variable in probability theory. Math expectation of a random variable x denoted M(x).

Mathematical expectation (Population mean) is

Mat waiting is

Mat waiting is in probability theory, the weighted average of all possible values ​​that this random variable can take.

Mat waiting is the sum of the products of all possible values ​​of a random variable by the probabilities of these values.

Mathematical expectation (Population mean) is

Mat waiting is the average benefit from a particular decision, provided that such a decision can be considered in the framework of the theory of large numbers and a long distance.

Mat waiting is in the theory of gambling, the amount of winnings that a speculator can earn or lose, on average, for each bet. In the language of gambling speculators this is sometimes called the "advantage speculator” (if it is positive for the speculator) or “house edge” (if it is negative for the speculator).

Mathematical expectation (Population mean) is

Mat 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 expectation. Let us introduce the concept of a system of random variables. 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 distribution of random variables and, which take values ​​from the set and, is given by probabilities.

The term "mat. expectation” was introduced by Pierre Simon Marquis de Laplace (1795) and originated from the concept of “expected value of the payoff”, which first appeared in the 17th century in the theory of gambling in the works of Blaise Pascal and Christian Huygens. However, the first complete theoretical understanding and evaluation 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.

Math expectation of a discrete random variable is the sum of the products of its possible values ​​and their corresponding probabilities. Sometimes mat. the expectation is called the weighted average, since it is approximately equal to the arithmetic mean of the observed values ​​of the random variable over a large number of experiments. From the definition of the expectation mat, it follows that its value is not less than the smallest possible value of a random variable and not more than the largest. Math expectation of a random variable is a non-random (constant) variable.

Math expectation has a simple physical meaning: if a unit mass is placed on a straight line, placing some mass at some points (for discrete distribution), or “smearing” it with a certain density (for an absolutely continuous distribution), then the point corresponding to the expectation will be the coordinate of the “center of gravity” of the straight line.

The average value of a random variable is a certain number, which is, as it were, its “representative” and replaces it in rough approximate calculations. When we say: “the average lamp operation time is 100 hours” or “the average point of impact is shifted relative to the target by 2 m to the right”, we indicate by this a certain numerical characteristic of a random variable that describes its location on the numerical axis, i.e. position description.

Of the characteristics of the situation in the theory of probability, the most important role is played by the expectation of a random variable, which is sometimes called simply the average value of a random variable.

Consider a random variable X, which has possible values x1, x2, …, xn with probabilities p1, p2, …, pn. We need to characterize by some number the position of the values ​​of the random variable on the x-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 mean of the random variable X, which we will denote M|X|:

This weighted average is called the mat expectation of the random variable. Thus, we introduced in consideration one of the most important concepts of probability theory - the concept of mat. 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. expectation of a random variable X due to a peculiar dependence with the arithmetic mean of the observed values ​​of a random variable with 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 mat. waiting. From the presence of a relationship between frequency and probability, one can deduce as a consequence the existence of a similar relationship between the arithmetic mean and mathematical expectation. Indeed, consider a random variable X, characterized by a series of distributions:

Let it be produced N independent experiments, in each of which the value X takes on a certain value. Suppose 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 X, which, in contrast to the expectation mats M|X| we will denote M*|X|:

With an increase in the number of experiments N frequencies pi will approach (converge in probability) the corresponding probabilities. Therefore, 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 expectation. The relation formulated above between the arithmetic mean and the mat. 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 certain 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 value. 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 not random" and, stabilizing, approaches a constant value - mat. waiting.

The property of stability of averages for a large number of experiments is easy to verify experimentally. For example, weighing any body in the laboratory on accurate scales, as a result of weighing we get a new value each time; to reduce the error of observation, 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 it 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 make examples of such random variables for which mat. there is no expectation, since the corresponding sum or integral diverges. However, for practice, such cases are not of significant interest. Usually, the random variables we deal with have a limited range of possible values ​​and, of course, have a mat expectation.

In addition to the most important of the characteristics of the position of a random variable - the expectation value - other characteristics of the position are sometimes used in practice, in particular, the mode and median of the random variable.

The mode of a random variable is its most probable value. The term "most likely 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 said to be "polymodal".

Sometimes there are distributions that have in the middle not a maximum, but a minimum. Such distributions are called "antimodal".

In the general case, the mode and expectation 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 the center of symmetry of the distribution.

Another characteristic of the position 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 as well. Geometrically, the median is the abscissa of the point at which the area bounded by the distribution curve is bisected.

In the case of a symmetric modal distribution, the median coincides with the mat. expectation and fashion.

Math expectation is an average value, random variable - a numerical characteristic of the probability distribution of a random variable. In the most general way, the mat 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. expectation can also be calculated as the Lebesgue integral of X by probability distribution px quantities X:

In a natural way, one can 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 are defined by many numerical and functional characteristics of the distribution (as the mathematical expectation of the corresponding functions of a random variable), for example, generating function, characteristic function, moments of any order, in particular variance, covariance.

Mathematical expectation (Population mean) is

Math 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. From other characteristics of the location, with the help of which the distribution is described in general terms - medians, modes, expectation differs in the greater value that it and the corresponding scattering characteristic - variance - have in the limit theorems of probability theory. With the greatest completeness, the meaning of expectation mats is revealed by the law of large numbers (Chebyshev's inequality) and the strengthened law of large numbers.

Mathematical expectation (Population mean) is

Mathematical 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 in a die roll 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 return (or loss) from each of the risky operations?

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 wins, the prize will be 300 rubles, and any ticket - 100 rubles. With an infinite number of participations, this is what happens. In three-quarters of the 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 an average of 100 rubles, for one - an average of 25 rubles. In total, the average rate of our ruin will be 25 rubles per ticket.

We throw a dice. If it's 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 take the stupid arithmetic mean and get 3.5. Since this is AVERAGE, there is no need to be indignant that no particular throw 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 take a look at the picture just above. On the left is a table of the distribution of a random variable. The value of X can take one of n possible values ​​(given in the top row). There can be no other values. Under each possible value, its probability is signed below. On the right is a formula, where M(X) is called mat. waiting. The meaning of this value is that with a large number of trials (with a large sample), the average value will tend to this very expectation.

Let's go back to the same playing cube. Mat. the expectation of the number of points when throwing is 3.5 (calculate yourself using the formula if you don’t believe it). Let's say you threw it a couple of times. 4 and 6 fell out. On average, it turned out 5, that is, far from 3.5. They threw it again, 3 fell out, 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 if the average is not exactly 3.5, then it will be close to that.

Let's count mat. waiting for the above described lottery. The table will look like this:

Then the expectation checkmate will be, as we have established above.:

Another thing is that it is also "on the fingers", without a formula, it would be difficult if there were more options. Well, let's say there were 75% losing tickets, 20% winning tickets, and 5% winning tickets.

Now some properties of expectation mat.

Mat. the wait is linear. It's easy to prove it:

The constant multiplier is allowed to be taken out of the checkmate sign. expectations, that is:

This is a special case of the linearity property of expectation mats.

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 mathematical expectations of random variables.

Let X, Y be independent random variables, then:

This is also easy to prove) XY itself is a random variable, while if the initial values ​​could take n and m values, respectively, then XY can take nm values. each of the values ​​is calculated based on the fact that the probabilities of independent events are multiplied. As a result, we get this:

Mathematical expectation of a continuous random variable

Continuous random variables have such a characteristic as the distribution density (probability density). It, in fact, characterizes the situation that a random variable takes some values ​​from the set of real numbers more often, some - less often. For example, consider this chart:

Here X- actually a random variable, f(x)- distribution density. Judging by this graph, during the experiments, the value X will often be a number close to zero. chances to exceed 3 or be less -3 rather purely theoretical.

If the distribution density is known, then the expectation mat is searched as follows:

Let, for example, there is a uniform distribution:

Let's find a mat. expectation:

This is quite consistent with the intuitive understanding. Let's say if we get a lot of 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 expectation mats - linearity, etc., applicable for discrete random variables, apply here as well.

The relationship of mathematical expectation with other statistical indicators

AT statistical analysis, along with mat expectation, there is a system of interdependent indicators that reflect the homogeneity of phenomena and stability processes. Often, variation indicators do not have 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 connected with the mat. waiting. This parameter is actively used in other types of statistical analysis (hypothesis testing, analysis of cause-and-effect relationships, etc.). Like the mean linear deviation, the variance also reflects the measure of spread data around the average.

It is useful to translate the language of signs into the language of words. It turns out that the variance 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 up and then divided by the number of values ​​in this population. Difference between a single value and the average reflects the measure of deviation. It is squared to ensure that all deviations become exclusively positive numbers and to avoid mutual cancellation of positive and negative deviations when they are summed. Then, given the squared deviations, we simply calculate the arithmetic mean. Average - squared - deviations. Deviations are squared, and the average is considered. The answer to the magic word "dispersion" is just three words.

However, in its pure form, such as, for example, the arithmetic mean, or , the dispersion is not used. It is rather an auxiliary and intermediate indicator that is used for other types of statistical analysis. She doesn't even have a normal unit of measure. Judging by the formula, this is the square of the original data unit.

Mathematical expectation (Population mean) is

Let's 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 mean value related to the distribution function?

Or we will roll the dice a large number of times. The number of points that will fall out on the die during each throw is a random variable and can take any natural values ​​from 1 to 6. N it tends to a very specific number - mat. expectation Mx. In this case, Mx = 3.5.

How did this value come about? Let in N trials n1 once 1 point is dropped, n2 times - 2 points and so on. Then the number of outcomes in which one point fell:

Similarly for the outcomes when 2, 3, 4, 5 and 6 points fell out.

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 on the values ​​x1, x2,..., xk with probabilities p1, p2,..., pk.

The mat expectation Mx of a random variable x is:

Math expectation is not always a reasonable estimate of some random variable. So, to estimate the average wage, it is more reasonable to use the concept of the median, that is, such a value that the number of people who receive less than the median salary and large, match.

The probability p1 that the random variable x is less than x1/2 and the probability p2 that the random variable x is greater than x1/2 are the same and equal to 1/2. The median is not uniquely determined 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 is grouped around the mean, and a large standard deviation indicates that the initial data is far from it. The standard deviation is equal to the square root of a quantity called the variance. It is the average of the sum of the squared differences of the initial data deviating from the mean. 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 variance and standard deviation of a random variable:

Variation- fluctuation, variability of the value of the attribute in units of the population. Separate numerical values ​​of a feature that occur in the studied population are called value variants. The insufficiency of the average value for a complete characterization of the population makes it necessary to supplement the average values ​​with indicators that make it possible to assess the typicality of these averages by measuring the fluctuation (variation) of the trait under study. The coefficient of variation is calculated by the formula:

Span variation(R) is the difference between the maximum and minimum values ​​of the trait in the studied population. This indicator gives the most general idea of ​​the fluctuation of the trait under study, as it shows difference only between the limit values ​​of the variants. The dependence on the extreme values ​​of the attribute gives the range of variation an unstable, random character.

Average linear deviation is the arithmetic mean of the absolute (modulo) deviations of all values ​​of the analyzed population from their average value:

Mathematical expectation in gambling theory

Mat waiting is the average amount of money a gambling speculator can win or lose on a given bet. This is a very significant concept for a speculator, because it is fundamental to the assessment of most gaming situations. Mate expectation is also the best tool for analyzing basic card layouts and game situations.

Let's say you're playing coin with a friend, making an equal $1 bet each time, no matter what comes up. Tails - you won, heads - you lost. The chances of it coming up tails are one to one and you are betting $1 to $1. Thus, your checkmate expectation is zero, because mathematically speaking, you can't know if you'll lead or lose after two rolls or after 200.

Your hourly gain is zero. Hourly payout is the amount of money you expect to win in an hour. You can flip a coin 500 times within an hour, but you won't win or lose because your odds are neither positive nor negative. If you look, from the point of view of a serious speculator, such a system of rates is not bad. But it's just a waste of time.

But suppose someone wants to bet $2 against your $1 in 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 the first and lose $1, bet the second and win $2. You've bet $1 twice and are ahead by $1. So each of your one dollar bets gave you 50 cents.

If the coin falls 500 times in one hour, your hourly gain will be already $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 win. Note that the expected value, which is the amount you win on average on a single bet, is 50 cents. You won $250 by betting a dollar 500 times, which equals 50 cents of your bet.

Mathematical expectation (Population mean) is

Mat. expectation has nothing to do with short-term results. Your opponent, who decided to bet $2 against you, could beat you on the first ten tosses in a row, but you, with a 2-to-1 betting advantage, all else being equal, make 50 cents on every $1 bet under any circumstances. It doesn't matter if you win or lose one bet or several bets, but only on the condition that you have enough cash to easily compensate for 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 expected values ​​in individual rolls.

Every time you make a best bet (a bet that can be profitable in the long run) when the odds are in your favor, you are bound to win something on it, whether you lose it or not in a given hand. Conversely, if you made a worse bet (a bet that is unprofitable in the long run) when the odds are not in your favor, you lose something, whether you win or lose the hand.

Mathematical expectation (Population mean) is

You bet with the best outcome if your expectation is positive, and it is positive if the odds are in your favor. By betting with the worst outcome, you have a negative expectation, which happens when the odds are against you. Serious speculators bet only with the best outcome, with the worst - they fold. What does the odds in your favor mean? You may end up winning more than the actual odds bring. The real odds of hitting tails are 1 to 1, but you get 2 to 1 due to the betting 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. expectations. The friend writes down the numbers from one to five and bets $5 against your $1 that you won't pick the number. Do you agree to such a bet? What is the expectation here?

On average, you'll be wrong four times. Based on this, the odds against you guessing the number will be 4 to 1. The odds are that you will lose a dollar in one attempt. However, you win 5 to 1, with the possibility of losing 4 to 1. Therefore, 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 four times $1 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 is going to win more than he bets, as in the example above, is catching the odds. Conversely, he ruins the chances when he expects to win less than he bets. The betting speculator can have either positive or negative expectation depending on whether he is catching or ruining 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 four times $10 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 $30 once, which is profit at $10. These examples show that the first bet is bad and the second is good.

Mat. expectation is the center of any game situation. When a bookmaker encourages football fans to bet $11 to win $10, they have a positive expectation of 50 cents for every $10. If the casino pays out even money from the Craps pass line, then the house's positive expectation is approximately $1.40 for every $100; this game is structured so that everyone who bets on this line loses 50.7% on average and wins 49.3% of the time. Undoubtedly, it is this seemingly minimal positive expectation that brings huge profits to casino owners around the world. As Vegas World casino owner Bob Stupak remarked, “One thousandth percent negative probability over a long enough distance will bankrupt the richest man in the world.

Mathematical expectation when playing poker

The game of Poker is the most illustrative and illustrative example in terms of the use of the theory and properties of the waiting mat.

Mat. Expected Value in Poker - the average benefit from a particular decision, provided that such a decision can be considered in the framework of the theory of large numbers and a long distance. Successful poker is about always accepting moves with a positive mathematical expectation.

Mathematical expectation (Population mean) is

Mathematical meaning. expectation when playing poker lies in the fact that we often encounter random variables when making a decision (we do not know which cards the opponent has in his hand, which cards will come on subsequent rounds trade). We must consider each of the solutions from the point of view of the theory of large numbers, which says that with a sufficiently large sample, the average value of a random variable will tend to its mean.

Among the particular formulas for calculating expectation mats, the following is most applicable in poker:

When playing poker mat. expectation can be calculated for both bets and calls. In the first case, fold equity should be taken into account, in the second, the pot's own odds. When evaluating mat. expectation of this or that move, it should be remembered that the 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 lose) for each risk you take. Casinos earn money because checkmate expectation from all the games that are practiced in them is in favor of the casino. With a sufficiently long series of games, it can be expected that the client will lose his money because the "probability" is 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 period of 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 in terms of checkmate. You can assume that a certain move is profitable, but in some cases it may not be the best one, because another move is more profitable. Let's say you hit a full house in five card draw poker. Your opponent bets. You know that if you up the ante, he will call. So raising looks like the best tactic. But if you do raise the bet, the remaining two speculators will definitely fold. But if you call the bet, you will be completely sure that the other two speculators after you will do the same. When you raise the bet, you get one unit, and simply by calling - two. So calling gives you a higher positive expected value and is the best tactic.

Mat. waiting can also give an idea of ​​which poker tactics are less profitable and which are more profitable. For example, if you play a particular hand and you think your average loss is 75 cents including the antes, 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 mat. expectation is that it gives you a sense of peace of mind whether you won the bet or not: if you made a good bet or folded in time, you will know that you have made or saved a certain amount of money that the weaker speculator could not save. It's much harder to fold if you're frustrated that your opponent has a better hand on the draw. With all this, what you save by not playing, instead of betting, is added to your winnings per night or per month.

Just remember that if you switched hands, your opponent would call you, and as you'll see in the Fundamental Theorem of Poker article, this is just one of your advantages. You should rejoice when this happens. You can even learn to enjoy a lost hand, because you know that other speculators in your place would lose much more.

As mentioned in the coin game example at the beginning, the hourly profit ratio is related to mat expectation, and this concept is especially important for professional speculators. When you are going to play poker, you must 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 mathematical calculations. For example, if you are playing draw lowball and you see three players bet $10 and then draw two cards, which is a very bad tactic, you can calculate for yourself that every time they bet $10 they lose about $2. Each of them does this eight times an hour, which means that all three lose about $48 per hour. You are one of the remaining four speculators, which are approximately equal, so these four speculators (and you among them) have to share $48, and each will make a profit of $12 per hour. Your hourly rate in this case is simply 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 profit of the speculator is the sum of his mathematical expectations in separate distributions. The more 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 prioritize a game that can maximize your positive expectation or negate your negative one so that you can maximize your hourly gain.

Positive mathematical expectation in game strategy

If you know how to count cards, you may have an advantage over the casino if they don't notice and kick you out. Casinos love drunk speculators and hate card counters. The advantage will allow you to win more times than you lose over time. Good money management using checkmate calculations can help you get more out of your edge and cut your losses. Without an advantage, you're better off giving the money to charity. In the game on the stock exchange, the advantage is given by the system of the game, which creates more profit than loss, the difference prices and commissions. None capital Management won't save a bad gaming system.

A positive expectation is defined by a value greater than zero. The larger this number, the stronger the statistical expectation. If the value is less than zero, then the expectation will also be negative. The greater the modulus of a negative value, the worse the situation. If the result is zero, then the expectation is breakeven. You can only win when you have a positive mathematical expectation, a reasonable game system. Playing on intuition leads to disaster.

Mathematical expectation and

Math expectation is a fairly widely demanded and popular statistical indicator in the implementation of exchange trading on financial markets. markets. First of all, this parameter is used to analyze the success trade. It is not difficult to guess that the larger this value, the more reason to consider the trade under study successful. Of course, analysis work trader cannot be made only with the help of this parameter. However, the calculated value in conjunction with other methods of assessing the quality work, can significantly improve the accuracy of the analysis.

Mat expectation is often calculated in trading account monitoring services, which allows you to quickly evaluate the work done on the deposit. As exceptions, we can cite strategies that use the “overstaying” of losing trades. Trader luck may accompany him for some time, and therefore, there may be no losses at all in his work. In this case, it will not be possible to navigate only by the expectation, because the risks used in the work will not be taken into account.

In trading on market mat expectation is most often used when predicting the profitability of a trading strategy or when forecasting income trader based on the statistics of his previous bidding.

Mathematical expectation (Population mean) is

In relation to money management, it is very important to understand that when making trades with a negative expectation, there is no scheme management money, which can definitely bring high profits. If you continue to play stock exchange under these conditions, regardless of the method management money, you will lose your entire account, no matter how big it was at the beginning.

This axiom is not only true for negative expectation games or trades, it is also true for even odds games. Therefore, the only case where you have a chance to benefit in the long run is when making deals with a positive mathematical expectation.

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; what matters is whether it is positive or negative. Therefore, before considering management issues capital you must find a game with a positive expectation.

If you don't have that game, then no amount of money management in the world will save you. On the other hand, if you have a positive expectation, then it is possible, through proper money management, to 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 based on one contract is. If you have a system that wins $10 per contract on a single trade (after commissions and slippage), management techniques can be used capital in a way to make it more profitable than a system that shows an average profit of $1,000 per trade (after fees and slippage).

What matters is not how profitable the system was, but how certain it can be said that the system will show at least a minimal profit in the future. Therefore, the most important preparation that can be made is to make sure that the system shows 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 want to build a fairly primitive and simple system that will constantly bring a small profit in almost any market. Again, it's important that you understand that it doesn't matter how profitable a system is, as long as it's profitable. that you earn in trading will be earned through efficient money management.

Mathematical expectation (Population mean) is

A trading system is simply a tool that gives you a positive mathematical expectation so that money management can be used. Systems that work (show at least a minimal profit) 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 technical traders is that they spend too much time and effort optimizing the various rules and parameters of a 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 the minimum profit.

Knowing that capital Management- this is just a number game that requires the use of positive expectations, the trader can stop looking for the "holy grail" of trading on the exchange. Instead, he can start testing his trading method, find out how logically this method is, whether it gives positive expectations. Proper money management methods applied to any, even very mediocre trading methods, will do the rest of the work.

For any trader to be successful in his work, he needs to solve the 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 the opportunity to earn money is as often as possible; Achieve a stable positive result of your operations.

And here, for us, working traders, checkmate can be a good help. expectation. This term in the theory of probability is one of the key. With it, you can give an average estimate of some random value. Math expectation of a random variable is similar to the center of gravity, if we imagine all possible probabilities as points with different masses.

In relation to a trading strategy, to evaluate its effectiveness, the expectation of profit (or loss) is most often used. This parameter is defined as the sum of the products of given profit and loss levels and the probability of their occurrence. For example, the developed trading strategy assumes that 37% of all operations will bring profit, and the rest - 63% - will be unprofitable. At the same time, the average income from a successful transaction will be 7 dollars, and the average loss will be equal to 1.4 dollars. Let's calculate mat. expectation of trading on such a system:

What does this number mean? It says that, following the rules of this system, on average, we will receive 1.708 dollars from each closed transaction. Since the resulting efficiency score is greater than zero, such a system can be used for real work. If, as a result of the calculation of the mat, 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 trade can also be expressed as a relative value in the form of %. For example:

Percentage of income per 1 transaction - 5%;

The percentage of successful trading operations - 62%;

Percentage of loss per 1 trade - 3%;

The percentage of unsuccessful transactions - 38%;

In this case, mat. expectation will be:

That is, the average transaction will bring 1.96%.

It is possible to develop a system that, despite the predominance of losing trades, will give 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. Suppose that each transaction averages only 0.5 dollars, but what if the system assumes 1000 transactions per year? This will be a very serious amount in a relatively short time. It logically follows from this that another hallmark of a good trading system can be considered a short holding period.

Sources and links

dic.academic.ru - academic online dictionary

mathematics.ru - educational site on mathematics

nsu.ru - educational website of Novosibirsk State University

webmath.ru - an educational portal for students, applicants and schoolchildren.

exponenta.ru educational mathematical site

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

unfx.ru - Forex at UNFX: training, trading signals, trust management

- - mathematical expectation One of the numerical characteristics of a random variable, often called its theoretical average. For a discrete random variable X, mathematical ... ... Technical Translator's Handbook

EXPECTED VALUE- (expected value) The average value of the distribution of the economic variable that it can take. If pt is the price of the good at time t, its mathematical expectation is denoted by Ept. To indicate the point in time to which ... ... Economic dictionary

Expected value- the average value of a random variable. The mathematical expectation is a deterministic quantity. The arithmetic mean of realizations of a random variable is an estimate of the mathematical expectation. Average… … The official terminology is the (mean value) of a random variable a numerical characteristic of a random variable. If a random variable given on a probability space (see Probability theory), then its M. o. MX (or EX) is defined as the Lebesgue integral: where... Physical Encyclopedia

EXPECTED VALUE- a random variable is its numerical characteristic. If a random variable X has a distribution function F(x), then its M. o. will be: . If the distribution of X is discrete, then М.о.: , where x1, x2, ... are possible values ​​of the discrete random variable X; p1 ... Geological Encyclopedia

EXPECTED VALUE- English. expected value; German Erwartung mathematische. Stochastic mean or center of dispersion of a random variable. Antinazi. Encyclopedia of Sociology, 2009 ... Encyclopedia of sociology

Expected value- See also: Conditional expectation Mathematical expectation is the mean value of a random variable, the probability distribution of a random variable, is considered in probability theory. In English literature and in mathematical ... ... Wikipedia

Expected value- 1.14 Mathematical expectation E (X) where xi values ​​of a discrete random variable; p = P (X = xi); f(x) is the density of a continuous random variable * If this expression exists in the sense of absolute convergence Source ... Dictionary-reference book of terms of normative and technical documentation

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The concept of mathematical expectation can be considered using the example of throwing a dice. With each throw, the dropped points are recorded. Natural values ​​in the range 1 - 6 are used to express them.

After a certain number of throws, using simple calculations, you can find the arithmetic mean of the points that have fallen.

As well as dropping any of the range values, this value will be random.

And if you increase the number of throws several times? With a large number of throws, the arithmetic mean value of the points will approach a specific number, which in probability theory is called the mathematical expectation.

So, the mathematical expectation is understood as the average value of a random variable. This indicator can also be presented as a weighted sum of probable values.

This concept has several synonyms:

• mean;
• average value;
• central trend indicator;
• first moment.

In other words, it is nothing more than a number around which the values ​​of a random variable are distributed.

In various spheres of human activity, approaches to understanding the mathematical expectation will be somewhat different.

It can be viewed as:

• the average benefit received from the adoption of a decision, in the case when such a decision is considered from the point of view of the theory of large numbers;
• the possible amount of winning or losing (gambling theory), calculated on average for each of the bets. In slang, they sound like "player's advantage" (positive for the player) or "casino advantage" (negative for the player);
• percentage of profit received from winnings.

Mathematical expectation is not obligatory for absolutely all random variables. It is absent for those who have a discrepancy in the corresponding sum or integral.

Expectation Properties

Like any statistical parameter, mathematical expectation has the following properties:

Basic formulas for mathematical expectation

The calculation of the mathematical expectation can be performed both for random variables characterized by both continuity (formula A) and discreteness (formula B):

1. M(X)=∑i=1nxi⋅pi, where xi are the values ​​of the random variable, pi are the probabilities:
2. M(X)=∫+∞−∞f(x)⋅xdx, where f(x) is a given probability density.

Examples of calculating the mathematical expectation

Example A.

Is it possible to find out the average height of the gnomes in the fairy tale about Snow White. It is known that each of the 7 gnomes had a certain height: 1.25; 0.98; 1.05; 0.71; 0.56; 0.95 and 0.81 m.

The calculation algorithm is quite simple:

• find the sum of all values ​​of the growth indicator (random variable):
  1,25+0,98+1,05+0,71+0,56+0,95+ 0,81 = 6,31;
• The resulting amount is divided by the number of gnomes:
  6,31:7=0,90.

Thus, the average height of the gnomes in the fairy tale is 90 cm. In other words, this is the mathematical expectation of the growth of the gnomes.

Working formula - M (x) \u003d 4 0.2 + 6 0.3 + 10 0.5 \u003d 6

Practical implementation of mathematical expectation

The calculation of a statistical indicator of mathematical expectation is resorted to in various fields of practical activity. First of all, we are talking about the commercial sphere. After all, the introduction of this indicator by Huygens is connected with the determination of the chances that can be favorable, or, on the contrary, unfavorable, for some event.

This parameter is widely used for risk assessment, especially when it comes to financial investments.
So, in business, the calculation of mathematical expectation acts as a method for assessing risk when calculating prices.

Also, this indicator can be used when calculating the effectiveness of certain measures, for example, on labor protection. Thanks to it, you can calculate the probability of an event occurring.

Another area of ​​application of this parameter is management. It can also be calculated during product quality control. For example, using mat. expectations, you can calculate the possible number of manufacturing defective parts.

Mathematical expectation is also indispensable during the statistical processing of the results obtained in the course of scientific research. It also allows you to calculate the probability of a desired or undesirable outcome of an experiment or study, depending on the level of achievement of the goal. After all, its achievement can be associated with gain and benefit, and its non-achievement - as a loss or loss.

Using Mathematical Expectation in Forex

The practical application of this statistical parameter is possible when conducting transactions in the foreign exchange market. It can be used to analyze the success of trade transactions. Moreover, an increase in the value of expectation indicates an increase in their success.

It is also important to remember that the mathematical expectation should not be considered as the only statistical parameter used to analyze the performance of a trader. The use of several statistical parameters along with the average value increases the accuracy of the analysis at times.

This parameter has proven itself well in monitoring observations of trading accounts. Thanks to him, a quick assessment of the work carried out on the deposit account is carried out. In cases where the trader's activity is successful and he avoids losses, it is not recommended to use only the calculation of mathematical expectation. In these cases, risks are not taken into account, which reduces the effectiveness of the analysis.

Conducted studies of traders' tactics indicate that:

• the most effective are tactics based on random input;
• the least effective are tactics based on structured inputs.

In order to achieve positive results, it is equally important:

• money management tactics;
• exit strategies.

Using such an indicator as the mathematical expectation, we can assume what will be the profit or loss when investing 1 dollar. It is known that this indicator, calculated for all games practiced in the casino, is in favor of the institution. This is what allows you to make money. In the case of a long series of games, the probability of losing money by the client increases significantly.

The games of professional players are limited to small time periods, which increases the chance of winning and reduces the risk of losing. The same pattern is observed in the performance of investment operations.

An investor can earn a significant amount with a positive expectation and a large number of transactions in a short time period.

Expectancy can be thought of as the difference between the percentage of profit (PW) times the average profit (AW) and the probability of loss (PL) times the average loss (AL).

As an example, consider the following: position - 12.5 thousand dollars, portfolio - 100 thousand dollars, risk per deposit - 1%. The profitability of transactions is 40% of cases with an average profit of 20%. In the event of a loss, the average loss is 5%. Calculating the mathematical expectation for a trade gives a value of $625.