Mathematical expectation formula. Expectation and variance of a random variable Days expectation week

The concept of mathematical expectation can be considered using the example of throwing a die. With each throw, the dropped points are recorded. To express them, natural values ​​in the range 1 – 6 are used.

After a certain number of throws, using simple calculations, you can find the arithmetic average of the points rolled.

Just like the occurrence of any of the values ​​in the range, this value will be random.

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

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

This concept has several synonyms:

• average value;
• average value;
• indicator of central tendency;
• first moment.

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

IN various fields human activity approaches to understanding mathematical expectation will be somewhat different.

It can be considered as:

• the average benefit obtained from making a decision, when such a decision is considered from the point of view of large number theory;
• the possible amount of winning or losing (gambling theory), calculated on average for each bet. 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.

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

Properties of mathematical expectation

Like any statistical parameter, the 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 the given probability density.

Examples of calculating mathematical expectation

Example A.

Is it possible to find out the average height of the dwarfs in the fairy tale about Snow White. It is known that each of the 7 dwarves 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:

• we 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;
• Divide the resulting amount by the number of gnomes:
  6,31:7=0,90.

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

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

Practical implementation of mathematical expectation

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

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

This indicator can also be used to calculate the effectiveness of certain measures, for example, 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 defective parts produced.

The mathematical expectation also turns out to be irreplaceable when carrying out statistical processing of the results obtained during scientific research results. It 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 failure can be associated with loss or loss.

Using mathematical expectation in Forex

Practical use this statistical parameter is possible when conducting operations on the foreign exchange market. With its help, you can analyze the success of trade transactions. Moreover, an increase in the expectation value 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 a trader’s performance. The use of several statistical parameters along with the average value increases the accuracy of the analysis significantly.

This parameter has proven itself well in monitoring observations of trading accounts. Thanks to it, 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 exclusively 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 tactics are those based on random entry;
• The least effective are tactics based on structured inputs.

In achieving positive results, no less important are:

• money management tactics;
• exit strategies.

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

Games played by professional players are limited to short periods of time, which increases the likelihood of winning and reduces the risk of losing. The same pattern is observed when performing investment operations.

An investor can earn a significant amount by having positive expectations and making a large number of transactions in a short period of time.

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

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

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

Let a random variable take only probability values ​​which are respectively equal. Then the mathematical expectation of a random variable is determined by the equality

If a discrete random variable 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.

Comment. From the definition it follows that the mathematical expectation of a discrete random variable is a non-random (constant) quantity.

Definition of mathematical expectation in the general case

Let us determine the mathematical expectation of a random variable whose distribution is not necessarily discrete. Let's start with the case of non-negative random variables. The idea will be to approximate such random variables using discrete ones for which the mathematical expectation has already been determined, and set the mathematical expectation equal to the limit of the mathematical expectations of the discrete random variables that approximate it. By the way, this is a very useful general idea, which is that some characteristic is first determined for simple objects, and then for more complex objects it is determined by approximating them by simpler ones.

Lemma 1. Let there be an arbitrary non-negative random variable. Then there is a sequence of discrete random variables such that

Proof. Let us divide the semi-axis into equal length segments and determine

Then properties 1 and 2 easily follow from the definition of a random variable, and

Lemma 2. Let be a non-negative random variable and and two sequences of discrete random variables possessing properties 1-3 from Lemma 1. Then

Proof. Note that for non-negative random variables we allow

By virtue of Property 3, it is easy to see that there is a sequence of positive numbers such that

It follows that

Using the properties of mathematical expectations for discrete random variables, we obtain

Passing to the limit at we obtain the statement of Lemma 2.

Definition 1. Let be a non-negative random variable, - a sequence of discrete random variables that have properties 1-3 from Lemma 1. The mathematical expectation of a random variable is the number

Lemma 2 guarantees that it does not depend on the choice of approximating sequence.

Let now be an arbitrary random variable. Let's define

From the definition and it easily follows that

Definition 2. The mathematical expectation of an arbitrary random variable is the number

If at least one of the numbers on the right side of this equality is finite.

Properties of mathematical expectation

Property 1. The mathematical expectation of a constant value is equal to the constant itself:

Proof. We will consider a constant as a discrete random variable that has one possible value and takes it with probability, therefore,

Remark 1. Let us define the product of a constant variable by a discrete random variable as a discrete random whose possible values ​​are equal to the products of the constant by the possible values; the probabilities of possible values ​​are equal to the probabilities of the corresponding possible values. For example, if the probability of a possible value is equal then the probability that the value will take the value is also equal

Property 2. The constant factor can be taken out of the sign of the mathematical expectation:

Proof. Let the random variable be given by the probability distribution law:

Taking into account Remark 1, we write the distribution law of the random variable

Remark 2. Before moving on to the next property, we point out that two random variables are called independent if the distribution law of one of them does not depend on what possible values ​​the other variable took. Otherwise, the random variables are dependent. Several random variables are called mutually independent if the laws of distribution of any number of them do not depend on what possible values ​​the remaining variables took.

Remark 3. Let us define the product of independent random variables and as a random variable whose possible values ​​are equal to the products of each possible value by each possible value, the probabilities of the possible values ​​of the product are equal to the products of the probabilities of the possible values ​​of the factors. For example, if the probability of a possible value is, the probability of a possible value is then the probability of a possible value is

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

Proof. Let independent random variables be specified by their own probability distribution laws:

Let's compile all the values ​​that a random variable can take. To do this, let's multiply all possible values ​​by each possible value; As a result, we obtain and, taking into account Remark 3, we write the distribution law, assuming for simplicity that all possible values ​​of the product are different (if this is not the case, then the proof is carried out in a similar way):

The mathematical expectation is equal to the sum of the products of all possible values ​​and their probabilities:

Consequence. The mathematical expectation of the product of several mutually independent random variables is equal to the product of their mathematical expectations.

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

Proof. Let random variables and be specified by the following distribution laws:

Let's compile all possible values ​​of a quantity. To do this, we add each possible value to each possible value; we obtain. Let us assume for simplicity that these possible values ​​are different (if this is not the case, then the proof is carried out in a similar way), and we denote their probabilities, respectively, by and

The mathematical expectation of a value is equal to the sum of the products of possible values ​​and their probabilities:

Let us prove that an Event that will take on the value (the probability of this event is equal) entails an event that will take on the value or (the probability of this event by the addition theorem is equal), and vice versa. Hence it follows that the equalities are proved similarly

Substituting the right-hand sides of these equalities into relation (*), we obtain

or finally

Variance and standard deviation

In practice, it is often necessary to estimate the dispersion of possible values ​​of a random variable around its mean value. For example, in artillery it is important to know how closely the shells will fall near the target that is to be hit.

At first glance, it may seem that the easiest way to estimate dispersion is to calculate all possible deviations of a random variable and then find their average value. However, this path will not give anything, since the average value of the deviation, i.e. for any random variable is equal to zero. This property is explained by the fact that some possible deviations are positive, while others are negative; as a result of their mutual cancellation, the average deviation value is zero. These considerations indicate the advisability of replacing possible deviations absolute values or their squares. This is what they do in practice. True, in the case when possible deviations are replaced by absolute values, one has to operate with absolute values, which sometimes leads to serious difficulties. Therefore, most often they take a different path, i.e. calculate the average value of the squared deviation, which is called dispersion.

Expectation is the probability distribution of a random variable

Mathematical expectation, definition, mathematical expectation of discrete and continuous random variables, sample, conditional expectation, calculation, properties, problems, estimation of expectation, dispersion, distribution function, formulas, calculation examples

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Mathematical expectation is the definition

One of the most important concepts in mathematical statistics and probability theory, characterizing the distribution of values ​​or probabilities of a random variable. Typically expressed as a weighted average of all possible parameters of a random variable. Widely used in technical analysis, research number series, the study of continuous and long-term processes. It is important when assessing risks, forecasting price indicators when trading on financial markets, is used in the development of strategies and methods of gaming tactics in the theory of gambling.

Mathematical expectation is the average value of a random variable, the probability distribution of a random variable is considered in probability theory.

Mathematical expectation is a measure of the average value of a random variable in probability theory. Expectation of a random variable x denoted by M(x).

Mathematical expectation is

Mathematical expectation is in probability theory, a weighted average of all possible values ​​that a random variable can take.

Mathematical expectation is the sum of the products of all possible values ​​of a random variable and the probabilities of these values.

Mathematical expectation 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.

Mathematical expectation is in gambling theory, the amount of winnings a player can earn or lose, on average, for each bet. In gambling parlance, this is sometimes called the "player's edge" (if it is positive for the player) or the "house edge" (if it is negative for the player).

Mathematical expectation is the percentage of profit per win multiplied by the average profit, minus the probability of loss multiplied by the average loss.

Mathematical expectation of a random variable in mathematical theory

One of the important numerical characteristics a random variable is its mathematical expectation. 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, the joint distribution law of random variables and, which take values ​​from the set and, is given by probabilities.

The term “mathematical 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 the theory of gambling 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).

The distribution law of random numerical variables (distribution function and distribution series or probability density) completely describes 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 the mathematical expectation, variance, mode and median.

The mathematical expectation of a discrete random variable is the sum of the products of its possible values ​​and their corresponding probabilities. Sometimes the mathematical expectation is called a weighted average, since it is approximately equal to the arithmetic mean of the observed values ​​of the random variable at large number experiments. From the definition of mathematical expectation it follows that its value is no less than the smallest possible value of a random variable and no more than the largest. The mathematical expectation 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 mathematical expectation 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 a random variable on the x-axis, taking into account the fact 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 mathematical expectation of the random variable. Thus, we introduced into consideration one of the most important concepts of probability theory - the concept of mathematical expectation. The mathematical 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.

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 mathematical expectation. 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 mathematical expectation. The connection between the arithmetic mean and mathematical expectation formulated above constitutes 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 - the mathematical expectation.

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 - the mathematical expectation - does not exist for all random variables. It is possible to compose examples of such random variables for which the mathematical expectation does not exist, since 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 characteristics of the position of a random variable - the mathematical expectation - 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 mathematical expectation of a random variable do not coincide. In the particular case, when the distribution is symmetrical and modal (i.e. has a mode) and there is a mathematical 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 mathematical expectation and mode.

The mathematical expectation is the average value of a random variable - a numerical characteristic of the probability distribution of a random variable. In the most general way, the mathematical 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:

The mathematical expectation can also be calculated as the Lebesgue integral of X by probability distribution px quantities X:

The concept of a random variable with infinite mathematical expectation can be defined in a natural way. A typical example is the return times of some random walks.

With the help of mathematical expectation, many numerical and functional characteristics distributions (as the mathematical expectation of corresponding functions from a random variable), for example, generating function, characteristic function, moments of any order, in particular dispersion, covariance.

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

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 the price of 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 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 the mathematical expectation. 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 mathematical expectation.

Let's return again to the same playing cube. The mathematical expectation of the 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 mathematical expectation. 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 mathematical expectation for the lottery described above. The plate will look like this:

Then the mathematical 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 of mathematical expectation.

It's easy to prove:

The constant factor can be taken out as a sign of the mathematical expectation, that is:

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

Another consequence of the linearity of the mathematical expectation:

that is, the mathematical 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. The probability of 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.

Let, for example, there be a uniform distribution:

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 expectation - linearity, etc., applicable for discrete random variables, are also applicable here.

Relationship between mathematical expectation and other statistical indicators

In statistical analysis, along with the mathematical expectation, there is a system of interdependent indicators that reflect the homogeneity of phenomena and the stability of 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 of the data, which is a valuable statistical characteristic.

The degree of variability or stability of processes in statistical science can be measured using several indicators.

The most important indicator characterizing the variability of a random variable is Dispersion, which is most closely and directly related to the mathematical expectation. 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, variance also reflects the extent of the spread of data around the mean 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. The 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. Average - square - deviations. Deviations are squared and the average is calculated. Solution magic word“variance” is just three words.

However, in its pure form, such as the arithmetic mean, or index, 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.

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'll roll the dice a large number of once. 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 - mathematical expectation Mx. In this case Mx = 3.5.

How did you get this value? Let in N tests n1 once you get 1 point, 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 distribution law 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.

The mathematical expectation Mx of a random variable x is equal to:

The mathematical expectation is not always a reasonable estimate of some random variable. So, to estimate the average wages it is more reasonable to use the concept of median, that is, such a value that the number of people receiving a salary lower than the median and a larger one 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 under study are called variants of values. Insufficient average value for full characteristics 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 about the variability of the studied characteristic, since it shows the difference only between the limiting 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:

Mathematical expectation in gambling theory

Mathematical expectation is The average amount of money a gambler can win or lose on a given bet. This is a very important concept for the player because it is fundamental to the assessment of most gaming situations. Mathematical expectation is also the 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 means you lose. The odds are one to one that it will come up heads, so you bet $1 to $1. Thus, your mathematical expectation is 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. If you look at it, from the point of view of a serious player, 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 the first dollar and you will lose $1, bet the 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 dollars 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 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.

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 players 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's more complex example mathematical expectation. 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 player who is going 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 bettor can have either a positive or a 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 $10 four times and lose $30 once, for a profit of $10. These examples show that the first bet is bad, and the second is good.

Mathematical expectation 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 enormous profits to casino owners around the world. As Vegas World casino owner Bob Stupak noted, “a one-thousandth of one percent negative probability over a long enough distance will ruin 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 mathematical expectation.

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.

The mathematical meaning of the mathematical expectation when playing poker is that we often encounter random variables when making decisions (we don’t know what cards the opponent has in his hands, what cards will come in subsequent rounds of betting). 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 mathematical expectation.

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

When playing poker, the expected value 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 the mathematical expectation of a particular move, you should remember that a fold always has a zero expectation. Thus, discarding cards will always be a more profitable decision than any negative move.

Expectation tells you what you can expect (profit or loss) for every dollar you risk. Casinos make money because the mathematical expectation of all games played in them is in favor of the casino. With a long enough series of games, you can expect that the client will lose his money, since the “odds” are in favor of the casino. However, professional casino players limit their games to short periods of time, thereby stacking the odds in their favor. The same goes for investing. If your expectation is positive, you can earn more money, making many trades in a short period of 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 considered from the standpoint of mathematical expectation. 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 players will definitely fold. But if you call, you have full confidence that the other two players behind 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.

The mathematical expectation can also give an idea of ​​which poker tactics are less profitable and which 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 to understand the concept of expected value is that it gives you a sense of peace of mind 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 the weaker player could not save. It's much harder to fold if you're upset because your opponent drew a stronger hand. With all this, the money you save by not playing instead of betting is added to your winnings for the night or 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 can even learn to enjoy losing a hand because you know that other players in your position would have lost much more.

As mentioned in the coin game example at the beginning, the hourly rate of profit is interrelated with the mathematical expectation, and this concept is especially important for professional players. 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 players who are approximately equal, so these four players (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 players in an hour.

Over a long period of time, the player’s total winnings are 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 anticipation or negate your negative anticipation 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, as long as they don't notice and throw you out. Casinos love drunk players and don't tolerate card counting players. The advantage will allow you to win over time. larger number times than to lose. Good money management using expected value calculations can help you extract more profit from your edge and reduce 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 game system, which creates greater profits than losses, price differences and commissions. No amount of money management can 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 the mathematical 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 stock trading

Mathematical expectation is a fairly widely used and popular statistical indicator when carrying out exchange trading in financial markets. First of all, this parameter is used to analyze the success of trading. It is not difficult to guess that the higher this value, the more reasons to consider the trade being studied successful. Of course, analysis of a trader’s work cannot be carried out using this parameter alone. However, the calculated value, in combination with other methods of assessing the quality of work, can significantly increase the accuracy of the analysis.

The mathematical expectation 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. A trader may be lucky 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 market trading, the mathematical expectation is most often used when predicting the profitability of any trading strategy or when predicting a trader’s income based on statistical data from his previous trading.

With regard to money management, it is very important to understand that when making trades with negative expectations, there is no money management scheme that can definitely bring high profits. If you continue to play the stock market under these conditions, then regardless of how you manage your money, you will lose your entire account, no matter how large it was to begin with.

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 money management, you should find a game with positive expectation.

If you don't have that game, then all the money management in the world won't 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 money management techniques to make it more profitable than a system that averages $1,000 per trade (after deduction of 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 a trader can make 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 make in trading will be made through effective money management.

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 enough. The problem with most technically oriented traders is that they spend too much time and effort optimizing various rules and parameter values 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 money 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, mathematical expectation can be of great help. 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 mathematical 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 mathematical expectation of profit (or loss) is most often used to evaluate its effectiveness. This parameter is defined as the sum of the products of 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 transaction will be $7, and the average loss will be $1.4. Let's calculate the mathematical 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 estimate is greater than zero, such a system can be used for real work. If, as a result of the calculation, the mathematical expectation turns out to be negative, then this already indicates an average loss and such trading will lead to ruin.

The amount of profit per transaction can also be expressed as a relative value in the form of %. For example:

– percentage of income per 1 transaction - 5%;

– percentage of successful trading operations - 62%;

– percentage of loss per 1 transaction - 3%;

– percentage of unsuccessful transactions - 38%;

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, its profitability 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 – educational portal for students, applicants and schoolchildren.

exponenta.ru educational mathematical website

ru.tradimo.com – free online school trading

crypto.hut2.ru – multidisciplinary information resource

poker-wiki.ru – free encyclopedia poker

sernam.ru – Science Library selected natural science publications

reshim.su – website WE WILL SOLVE test coursework problems

unfx.ru – Forex on UNFX: training, trading signals, trust management

slovopedia.com – Big encyclopedic Dictionary Slovopedia

pokermansion.3dn.ru – Your guide in the world of poker

statanaliz.info – information blog “ Statistical analysis data"

forex-trader.rf – Forex-Trader portal

megafx.ru – current Forex analytics

fx-by.com – everything for a trader

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 is approximately equal to the average value of the random variable.

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

If a random variable is characterized by a finite distribution series:

X	x 1	x 2	x 3	…	x n
R	p 1	p 2	p 3	…	r p

then the mathematical expectation M(X) determined by the formula:

The mathematical expectation of a continuous random variable is determined by the equality:

where is the probability density of the random variable X.

Example 4.7. Find the mathematical expectation of the number of points that appear when throwing a dice.

Solution:

Random value X takes the values ​​1, 2, 3, 4, 5, 6. Let’s create the law of its distribution:

X
R

Then the mathematical expectation is:

Properties of mathematical expectation:

1. The mathematical expectation of a constant value is equal to the constant itself:

M (S) = S.

2. The constant factor can be taken out of the mathematical 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).

Example 4.8. Independent random variables X And Y are given by the following distribution laws:

X					Y
R	0,6	0,1	0,3		R	0,8	0,2

Find the mathematical expectation of the random variable XY.

Solution.

Let's find the mathematical expectations of each of these quantities:

Random variables X And Y independent, therefore the required mathematical expectation is:

M(XY) = M(X)M(Y)=

Consequence. The mathematical expectation of the product of several mutually independent random variables is equal to the product of their mathematical expectations.

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).

Consequence. The mathematical expectation of the sum of several random variables is equal to the sum of the mathematical expectations of the terms.

Example 4.9. 3 shots are fired with probabilities of hitting the target equal to p 1 = 0,4; p2= 0.3 and p 3= 0.6. Find the mathematical expectation of the total number of hits.

Solution.

The number of hits on the first shot is a random variable X 1, which can only take two values: 1 (hit) with probability p 1= 0.4 and 0 (miss) with probability q 1 = 1 – 0,4 = 0,6.

The mathematical expectation of the number of hits on the first shot is equal to the probability of a hit:

Similarly, we find the mathematical expectations of the number of hits for the second and third shots:

M(X 2)= 0.3 and M(X 3)= 0,6.

Total number hits is also a random variable consisting of the sum of hits in each of the three shots:

X = X 1 + X 2 + X 3.

The required mathematical expectation X We find it using the theorem on the mathematical expectation of the sum.

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 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.