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 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) \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 profit, 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.

The mathematical expectation is the probability distribution of a random variable

Mathematical expectation, definition, mathematical expectation of discrete and continuous random variables, selective, conditional expectation, calculation, properties, tasks, estimation of expectation, variance, distribution function, formulas, calculation examples

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The 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. 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 the theory of gambling.

The mathematical expectation is the mean value of a random variable, the probability distribution of a random variable is considered in probability theory.

The mathematical expectation is measure of the mean value of a random variable in probability theory. Mathematical expectation of a random variable x denoted M(x).

The mathematical expectation is

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

The mathematical expectation is the sum of the products of all possible values ​​of a random variable by the probabilities of these values.

The mathematical expectation 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.

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

The mathematical expectation is Percentage of profit per win multiplied by average profit minus probability of loss multiplied by average loss.

Mathematical expectation of a random variable in mathematical theory

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

The term "expectation" was introduced by Pierre Simon Marquis de Laplace (1795) and originated from the concept of "expected value of 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).

The distribution law of random numerical variables (the distribution function and the 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 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 the weighted average, since it is approximately equal to the arithmetic mean of the observed values ​​of a random variable over a large number of experiments. From the definition of mathematical expectation, it follows that its value is not less than the smallest possible value of a random variable and not 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 a unit mass is placed on a straight line, placing some 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" straight.

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 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 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, 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 mean of the random variable X, which we will denote M|X|:

This weighted average is called the mathematical expectation of the random variable. Thus, we introduced in 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 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) its mathematical expectation. 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 mathematical expectation 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 mathematical expectation. The connection between the arithmetic mean and the 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 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 - the mathematical expectation.

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 - the mathematical expectation - does not exist for all random variables. It is possible to make examples of such random variables for which the mathematical expectation does not exist, since the corresponding sum or integral diverges. However, for practice, such cases are not of significant interest. Usually, the random variables we are dealing with have a limited range of possible values ​​and, of course, have an expectation.

In addition to the most important of the characteristics of the position of a random variable - the mathematical expectation, other position characteristics 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 the mathematical expectation of a random variable do not coincide. In a particular case, when the distribution is symmetric and modal (i.e. has a mode) and there is a mathematical 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 mean and the mode.

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:

In a natural way, one can define the concept of a random variable with infinite mathematical expectation. A typical example is the return times in some random walks.

With the help of mathematical expectation, many numerical and functional characteristics of the distribution are determined (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 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, the mathematical expectation differs in the greater value that it and the corresponding scattering characteristic - dispersion - have in the limit theorems of probability theory. With the greatest completeness, the meaning of mathematical expectation is revealed by the law of large numbers (Chebyshev's inequality) and the strengthened law of large numbers.

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 the price of any ticket will be 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 the mathematical expectation. 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 mathematical expectation.

Let's go back to the same playing cube. The mathematical expectation of the number of points in a throw 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 mathematical expectation. 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 calculate the mathematical expectation for the above described lottery. The table will look like this:

Then the mathematical expectation 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 mathematical expectation.

It's easy to prove it:

A constant multiplier can be taken out of the expectation sign, 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) XY itself is a random variable, while if the initial values ​​could take n and m values, respectively, then XY can take nm values. The probability of 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.

Let, for example, there is a uniform distribution:

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 mathematical expectation - linearity, etc., applicable for discrete random variables, are applicable here as well.

The relationship of mathematical expectation with other statistical indicators

In statistical analysis, along with mathematical expectation, there is a system of interdependent indicators that reflect the homogeneity of phenomena and the stability of 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 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 mean linear deviation, the variance also reflects the extent to which the data spread around the mean.

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. The difference between the individual value and the mean reflects the measure of the 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 - square - 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 index, 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.

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 - the mathematical 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 distribution law of the random variable x, that is, we know that the random variable x can take the values ​​x1, x2, ..., xk with probabilities p1, p2, ..., pk.

The mathematical expectation Mx of a random variable x is:

The mathematical 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 more, are the same.

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 variants of values. 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 the difference only between the extreme 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

The mathematical expectation is the average amount of money a gambler can win or lose on a given bet. This is a very significant concept for a player, because it is fundamental to the assessment of most game situations. Mathematical 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 win, heads - you lose. The chances of it coming up tails are one to one and you are betting $1 to $1. Thus, your mathematical 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 player, such a betting system 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 dollar 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 $1 250 times and won $2 250 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 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 keep betting the same way, then over a long period of time your winnings will come up to 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.

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 players only bet 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 of mathematical expectation. 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 player 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 bettor 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, 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, 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, “A one-thousandth of a 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 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 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.

The mathematical meaning of the mathematical expectation when playing poker is that we often encounter random variables when making a decision (we do not know which cards are in the opponent's hand, which cards will come on subsequent betting rounds). 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 mathematical expectation.

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

When playing poker, the mathematical 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 the mathematical expectation of a particular move, it should be remembered that a fold always has a zero mathematical 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 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, since the “probability” is in favor of the casino. However, professional casino players 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. The expectation is your percentage of profit per win times your average profit minus your probability of loss times your average loss.

Poker can also be considered in terms of mathematical expectation. 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 remaining two players will fold for sure. But if you call the bet, you will be completely sure that the other two players after you will do the same. When you raise the bet, you get one unit, and simply by calling you get two. So calling gives you a higher positive expected value and is 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 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 expected value is that it gives you a sense of peace of mind whether you win a bet or not: if you made a good bet or folded in time, you will know that you have earned or saved a certain amount of money, which a weaker player could not save. It's much harder to fold if you're frustrated that your opponent has a better hand on the draw. That said, the money you save by not playing, instead of betting, is added to your overnight or monthly winnings.

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 losing a hand, because you know that other players in your shoes would lose much more.

As discussed in the coin game example at the beginning, the hourly rate of return is related to the mathematical expectation, and this concept is especially important for professional players. 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 players, which are approximately equal, so these four players (and you among them) must share $48, and each will earn $12 per hour. Your hourly rate in this case is simply your share of the amount of money lost by three bad players per hour.

Over a long period of time, the total winnings of the player 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 drunken gamblers and can't stand counting cards. The advantage will allow you to win more times than you lose over time. Good money management using expectation calculations can help you capitalize on 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 losses, price differences and commissions. No amount of money management will 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 mathematical expectation will also be negative. The larger the modulus of a negative value, the worse the situation. If the result is zero, then the expectation is break even. You can only win when you have a positive mathematical expectation, a reasonable game system. Playing on intuition leads to disaster.

Mathematical expectation and stock trading

Mathematical expectation is a fairly widely demanded and popular statistical indicator in 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 larger this value, the more reason to consider the trade under study successful. Of course, the analysis of the work of a trader cannot be carried out only with the help of this parameter. 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. As exceptions, we can cite strategies that use the “overstaying” of losing trades. A trader may be lucky for some time, and therefore, in his work there may be no losses at all. 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 the market, mathematical expectation is most often used when predicting the profitability of a trading strategy or when predicting a trader's income based on the statistics of his previous trades.

In terms of money management, it is very important to understand that when making trades with negative expectation, there is no money management scheme that can definitely bring high profits. If you continue to play the exchange under these conditions, then regardless of how you manage your 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 money management, 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 fees and slippage), you can use money management techniques to make it more profitable than a system that shows an average profit of $1,000 per trade (after deduction of commissions 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 a trader can make 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. The money you earn in trading will be earned through effective money management.

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 a minimum profit.

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

Any trader for success in their work 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 the opportunity to earn money is as often as possible; Achieve a stable positive result of your operations.

And here, for us, working traders, mathematical expectation can provide a good help. This term in the theory of probability is one of the key. With it, you can give an average estimate of some random value. The mathematical expectation of a random variable is like 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 mathematical expectation of profit (or loss) is most often used. 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 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, and the average loss will be $1.4. Let's calculate the mathematical expectation of trading using the following 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, 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 trade 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%;

– loss percentage per 1 trade - 3%;

- the percentage of unsuccessful transactions - 38%;

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, its profitability will be comparable to bank interest. Let each operation bring in only 0.5 dollars on average, 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 is 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 SOLVE tasks control coursework

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

slovopedia.com - Big Encyclopedic Dictionary

pokermansion.3dn.ru - Your guide to the world of poker

statanaliz.info - informational blog "Statistical data analysis"

forex-trader.rf - portal Forex-Trader

megafx.ru - up-to-date Forex analytics

fx-by.com - everything for a trader

01.02.2018

Expected value. Just about the complex. The basics of trading.

When placing bets of any type, there is always a certain probability of profit and the risk of failure. The positive outcome of the transaction, and the risk of losing money are inextricably linked with the mathematical expectation. In this article, we will focus on these two aspects of trading in detail.

Expected value- with the number of samples or the number of its measurements (sometimes they say - the number of tests) tending to infinity.

The point is that a positive expected value leads to a positive (increasing profit) trade, while a zero or negative expected value means no trading at all.

To make it easier to understand this issue, let's consider the concept of mathematical expectation when playing roulette. The roulette example is very easy to understand.

Roulette- (The croupier launches the ball in the opposite direction of the rotation of the wheel, from the number on which the ball fell the previous time, which must fall into one of the numbered cells, making at least three full revolutions around the wheel.

Cells numbered from 1 to 36 are colored black and red. The numbers are not in order, although the colors of the cells strictly alternate, starting with 1 - red. The cell marked with the number 0 is colored green and is called zero.

Roulette is a game with a negative mathematical expectation. All because of the field zero. "0", which is neither black nor red.

Because (generally) if no change is applied, the player loses $1 for every 37 spins of the wheel (when betting $1 at a time), resulting in a linear loss of -2.7% that increases as the number of bets increases (average).

Of course, a player in the interval, for example, 1000 games, may have a series of victories, and a person may mistakenly believe that he can earn by beating the casino and a series of defeats. A series of victories in this case can increase the player's capital by a greater value than he had initially, in this case, if the player had $1000, after 10 games of $1 each, he should have an average of $973 left. But if in such a scenario the player has less or more money, we will call such a difference between the current capital variance. You can only make money playing roulette within the variance. If the player continues to follow this strategy, eventually the person will be left without money, and the casino will work.

The second example is the famous binary options. You are allowed to make a bet, with a successful outcome, you take as much as 90 percent on top of your bet, and if unsuccessful, you lose all 100. And then the owners of the BO just have to wait, the market and the negative checkmate expectation will do their job. And the time dispersion will give hope to the binary options trader that it is possible to earn money in this market. But this is temporary.

What is the advantage of cryptocurrency trading (as well as trading in the stock market)?

A person can create a system for himself. He himself can limit his risk, and try to take the maximum possible profit from the market. (Moreover, if the situation with the second is rather controversial, then the risk must be controlled very clearly.)

To understand in which direction your strategy is leading you, you need to keep statistics. The trader needs to know:

1. The number of your trades. The greater the number of trades for a given strategy, the more accurate the mathematical expectation will be.
2. The frequency of successful entries. (Probability) (R)
3. Your profit for each positive transaction.
4. Bias (winning ratio) (B)
5. Average size of your bet (stop order) (S)

Expectation (E) = B * R - (1 - B) = B * (1 + R) -1

To roughly find out your final earnings or loss on the account (EE), for example, at a distance of 1000 trades, we will use the formula.

Where N is the number of trades that we plan to execute.

For example, let's take the initial data:

stop loss - 30 dollars.

profit - 100 dollars.

Number of transactions 30

Mathematical expectation is negative only if the ratio of profitable and losing trades (R) is 20%/80% or worse. In other cases, it is positive.

Now let the profit be 150. Then the expectation will be negative at a ratio of 16%/84%. Or lower.

Conclusion.

What to do with it? Start keeping statistics if you haven't already. Check your trades, determine your checkmate expectation. Find something that can be improved (number of correct entries, adding profits, cutting losses)

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- the number of boys among 10 newborns.

It is quite clear that this number is not known in advance, and in the next ten children born there may be:

Or boys - one and only one of the listed options.

And, in order to keep in shape, a little physical education:

- long jump distance (in some units).

Even the master of sports is not able to predict it :)

However, what are your hypotheses?

2) Continuous random variable - takes all numeric values ​​from some finite or infinite range.

Note : abbreviations DSV and NSV are popular in educational literature

First, let's analyze a discrete random variable, then - continuous.

Distribution law of a discrete random variable

- This conformity between the possible values ​​of this quantity and their probabilities. Most often, the law is written in a table:

The term is quite common row distribution, but in some situations it sounds ambiguous, and therefore I will adhere to the "law".

And now very important point: since the random variable necessarily will accept one of the values, then the corresponding events form full group and the sum of the probabilities of their occurrence is equal to one:

or, if written folded:

So, for example, the law of the distribution of probabilities of points on a die has the following form:

No comments.

You may be under the impression that a discrete random variable can only take on "good" integer values. Let's dispel the illusion - they can be anything:

Example 1

Some game has the following payoff distribution law:

…probably you have been dreaming about such tasks for a long time :) Let me tell you a secret - me too. Especially after finishing work on field theory.

Decision: since a random variable can take only one of three values, the corresponding events form full group, which means that the sum of their probabilities is equal to one:

We expose the "partisan":

– thus, the probability of winning conventional units is 0.4.

Control: what you need to make sure.

Answer:

It is not uncommon when the distribution law needs to be compiled independently. For this use classical definition of probability, multiplication / addition theorems for event probabilities and other chips tervera:

Example 2

There are 50 lottery tickets in the box, 12 of which are winning, and 2 of them win 1000 rubles each, and the rest - 100 rubles each. Draw up a law of distribution of a random variable - the size of the winnings, if one ticket is randomly drawn from the box.

Decision: as you noticed, it is customary to place the values ​​of a random variable in ascending order. Therefore, we start with the smallest winnings, and namely rubles.

In total, there are 50 - 12 = 38 such tickets, and according to classical definition:
is the probability that a randomly drawn ticket will not win.

The rest of the cases are simple. The probability of winning rubles is:

Checking: - and this is a particularly pleasant moment of such tasks!

Answer: the required payoff distribution law:

The following task for an independent decision:

Example 3

The probability that the shooter will hit the target is . Make a distribution law for a random variable - the number of hits after 2 shots.

... I knew that you missed him :) We remember multiplication and addition theorems. Solution and answer at the end of the lesson.

The distribution law completely describes a random variable, but in practice it is useful (and sometimes more useful) to know only some of it. numerical characteristics .

Mathematical expectation of a discrete random variable

In simple terms, this average expected value with repeated testing. Let a random variable take values ​​with probabilities respectively. Then the mathematical expectation of this random variable is equal to sum of works all its values ​​by the corresponding probabilities:

or in folded form:

Let's calculate, for example, the mathematical expectation of a random variable - the number of points dropped on a dice:

Now let's recall our hypothetical game:

The question arises: is it even profitable to play this game? ... who has any impressions? So you can’t say “offhand”! But this question can be easily answered by calculating the mathematical expectation, in essence - weighted average probabilities of winning:

Thus, the mathematical expectation of this game losing.

Don't trust impressions - trust numbers!

Yes, here you can win 10 or even 20-30 times in a row, but in the long run we will inevitably be ruined. And I would not advise you to play such games :) Well, maybe only for fun.

From all of the above, it follows that the mathematical expectation is NOT a RANDOM value.

Creative task for independent research:

Example 4

Mr X plays European roulette according to the following system: he constantly bets 100 rubles on red. Compose the law of distribution of a random variable - its payoff. Calculate the mathematical expectation of winnings and round it up to kopecks. How much average does the player lose for every hundred bet?

Reference : European roulette contains 18 red, 18 black and 1 green sector ("zero"). In the event of a “red” falling out, the player is paid a double bet, otherwise it goes to the casino’s income

There are many other roulette systems for which you can create your own probability tables. But this is the case when we do not need any distribution laws and tables, because it is established for certain that the mathematical expectation of the player will be exactly the same. Only changes from system to system

As already known, the distribution law completely characterizes a random variable. However, the distribution law is often unknown and one has to limit oneself to lesser information. Sometimes it is even more profitable to use numbers that describe a random variable in total; such numbers are called numerical characteristics of a random variable. Mathematical expectation is one of the important numerical characteristics.

The mathematical expectation, as will be shown below, is approximately equal to the average value of the random variable. To solve many problems, it is enough to know the mathematical expectation. For example, if it is known that the mathematical expectation of the number of points scored by the first shooter is greater than that of the second, then the first shooter, on average, knocks out more points than the second, and therefore shoots better than the second. Although the mathematical expectation gives much less information about a random variable than the law of its distribution, but for solving problems like the one given and many others, knowledge of the mathematical expectation is sufficient.

§ 2. Mathematical expectation of a discrete random variable

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

Let the random variable X can only take values X 1 , X 2 , ..., X P , whose probabilities are respectively equal R 1 , R 2 , . . ., R P . Then the mathematical expectation M(X) random variable X is defined by the equality

M(X) = X 1 R 1 + X 2 R 2 + … + x n p n .

If a discrete random variable X takes on a countable set of possible values, then

M(X)=

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

Comment. It follows from the definition that the mathematical expectation of a discrete random variable is a non-random (constant) variable. We recommend that you remember this statement, as it is used repeatedly later on. Later it will be shown that the mathematical expectation of a continuous random variable is also a constant value.

Example 1 Find the mathematical expectation of a random variable X, knowing the law of its distribution:

Decision. The desired mathematical expectation is equal to the sum of the products of all possible values ​​of a random variable and their probabilities:

M(X)= 3* 0, 1+ 5* 0, 6+ 2* 0, 3= 3, 9.

Example 2 Find the mathematical expectation of the number of occurrences of an event BUT in one trial, if the probability of an event BUT is equal to R.

Decision. Random value X - number of occurrences of the event BUT in one test - can take only two values: X 1 = 1 (event BUT happened) with a probability R and X 2 = 0 (event BUT did not occur) with a probability q= 1 -R. The desired mathematical expectation

M(X)= 1* p+ 0* q= p

So, the mathematical expectation of the number of occurrences of an event in one trial is equal to the probability of this event. This result will be used below.

§ 3. Probabilistic meaning of mathematical expectation

Let produced P tests in which the random variable X accepted t 1 times value X 1 , t 2 times value X 2 ,...,m k times value x k , and t 1 + t 2 + …+t to = p. Then the sum of all values ​​taken X, is equal to

X 1 t 1 + X 2 t 2 + ... + X to t to .

Find the arithmetic mean of all values ​​accepted as a random variable, for which we divide the found sum by the total number of trials:

= (X 1 t 1 + X 2 t 2 + ... + X to t to)/P,

= X 1 (m 1 / n) + X 2 (m 2 / n) + ... + X to (t to /P). (*)

Noticing that the relationship m 1 / n- relative frequency W 1 values X 1 , m 2 / n - relative frequency W 2 values X 2 etc., we write the relation (*) as follows:

=X 1 W 1 + x 2 W 2 + .. . + X to W k . (**)

Let us assume that the number of trials is sufficiently large. Then the relative frequency is approximately equal to the probability of occurrence of the event (this will be proved in Chapter IX, § 6):

W 1 p 1 , W 2 p 2 , …, W k p k .

Replacing the relative frequencies in relation (**) with the corresponding probabilities, we obtain

x 1 p 1 + X 2 R 2 + … + X to R to .

The right side of this approximate equality is M(X). So,

M(X).

The probabilistic meaning of the result obtained is as follows: mathematical expectation is approximately equal to(the more accurate the greater the number of trials) the arithmetic mean of the observed values ​​of the random variable.

Remark 1. It is easy to see that the mathematical expectation is greater than the smallest and less than the largest possible values. In other words, on the number axis, the possible values ​​are located to the left and right of the expected value. In this sense, the expectation characterizes the location of the distribution and is therefore often referred to as distribution center.

This term is borrowed from mechanics: if the masses R 1 , R 2 , ..., R P located at points with abscissas x 1 , X 2 , ..., X n, and
then the abscissa of the center of gravity

x c =
.

Given that
= M (X) and
we get M(X)= x with .

So, the mathematical expectation is the abscissa of the center of gravity of a system of material points, the abscissas of which are equal to the possible values ​​of a random variable, and the masses are equal to their probabilities.

Remark 2. The origin of the term "expectation" is associated with the initial period of the emergence of probability theory (XVI-XVII centuries), when its scope was limited to gambling. The player was interested in the average value of the expected payoff, or, in other words, the mathematical expectation of the payoff.