The mathematical expectation is the probability distribution of a random variable. Mathematical expectation (Population mean) is

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? At large quantities 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 a theoretical point of view large numbers;
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):

M(X)=∑i=1nxi⋅pi, where xi are the values of the random variable, pi are the probabilities:
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're talking about 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

The practical application of this statistical parameter is possible when conducting transactions on the foreign exchange market. It can be used to analyze success trade deals. 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 with positive anticipation and execution. large quantity transactions over 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.

Mathematical expectation is the definition

Checkmate waiting is one of the most important concepts in mathematical statistics and probability theory, characterizing the distribution of values or probabilities random variable. Typically expressed as a weighted average of all possible parameters of a random variable. Widely used in technical analysis, the study of number series, and the study of continuous and time-consuming processes. It is important in assessing risks, predicting price indicators when trading on financial markets, and is used in developing strategies and methods of gaming tactics in gambling theories.

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

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

Mathematical expectation (Population mean) is

Checkmate waiting is

Checkmate waiting is in probability theory, a weighted average of all possible values that a random variable can take.

Checkmate waiting is the sum of the products of all possible values of a random variable and the probabilities of these values.

Mathematical expectation (Population mean) is

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

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

Mathematical expectation (Population mean) is

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Probability theory is a special branch of mathematics that is studied only by students of higher educational institutions. Do you like calculations and formulas? Aren't you scared by the prospects of getting acquainted with the normal distribution, ensemble entropy, mathematical expectation and dispersion of a discrete random variable? Then this subject will be very interesting to you. Let's take a look at a few of the most important basic concepts this branch of science.

Let's remember the basics

Even if you remember the simplest concepts of probability theory, do not neglect the first paragraphs of the article. The point is that without a clear understanding of the basics, you will not be able to work with the formulas discussed below.

So, some random event occurs, some experiment. As a result of the actions we take, we can get several outcomes - some of them occur more often, others less often. The probability of an event is the ratio of the number of actually obtained outcomes of one type to total number possible. Only knowing the classical definition of this concept can you begin to study the mathematical expectation and dispersion of continuous random variables.

Average

Back in school, during math lessons, you started working with the arithmetic mean. This concept is widely used in probability theory, and therefore cannot be ignored. The main thing for us at the moment is that we will encounter it in the formulas for the mathematical expectation and dispersion of a random variable.

We have a sequence of numbers and want to find the arithmetic mean. All that is required of us is to sum up everything available and divide by the number of elements in the sequence. Let us have numbers from 1 to 9. The sum of the elements will be equal to 45, and we will divide this value by 9. Answer: - 5.

Dispersion

Speaking scientific language, dispersion is the average square of deviations of the obtained characteristic values from the arithmetic mean. It is denoted by one capital Latin letter D. What is needed to calculate it? For each element of the sequence, we calculate the difference between the existing number and the arithmetic mean and square it. There will be exactly as many values as there can be outcomes for the event we are considering. Next, we sum up everything received and divide by the number of elements in the sequence. If we have five possible outcomes, then divide by five.

Dispersion also has properties that need to be remembered in order to be used when solving problems. For example, when increasing a random variable by X times, the variance increases by X squared times (i.e. X*X). She never happens less than zero and does not depend on shifting values by an equal value up or down. Additionally, for independent trials, the variance of the sum is equal to the sum of the variances.

Now we definitely need to consider examples of the variance of a discrete random variable and the mathematical expectation.

Let's say we ran 21 experiments and got 7 different outcomes. We observed each of them 1, 2, 2, 3, 4, 4 and 5 times, respectively. What will the variance be equal to?

First, let's calculate the arithmetic mean: the sum of the elements, of course, is 21. Divide it by 7, getting 3. Now subtract 3 from each number in the original sequence, square each value, and add the results together. The result is 12. Now all we have to do is divide the number by the number of elements, and, it would seem, that’s all. But there's a catch! Let's discuss it.

Dependence on the number of experiments

It turns out that when calculating variance, the denominator can contain one of two numbers: either N or N-1. Here N is the number of experiments performed or the number of elements in the sequence (which is essentially the same thing). What does this depend on?

If the number of tests is measured in hundreds, then we must put N in the denominator. If in units, then N-1. Scientists decided to draw the border quite symbolically: today it passes through the number 30. If we conducted less than 30 experiments, then we will divide the amount by N-1, and if more, then by N.

Task

Let's return to our example of solving the problem of variance and mathematical expectation. We got an intermediate number 12, which needed to be divided by N or N-1. Since we conducted 21 experiments, which is less than 30, we will choose the second option. So the answer is: the variance is 12 / 2 = 2.

Expected value

Let's move on to the second concept, which we must consider in this article. The mathematical expectation is the result of adding all possible outcomes multiplied by the corresponding probabilities. It is important to understand that the obtained value, as well as the result of calculating the variance, is obtained only once for the whole task, no matter how many outcomes are considered.

The formula for mathematical expectation is quite simple: we take the outcome, multiply it by its probability, add the same for the second, third result, etc. Everything related to this concept is not difficult to calculate. For example, the sum of the expected values is equal to the expected value of the sum. The same is true for the work. Not every quantity in probability theory allows you to perform such simple operations. Let's take the problem and calculate the meaning of two concepts we have studied at once. Besides, we were distracted by theory - it's time to practice.

One more example

We ran 50 trials and got 10 types of outcomes - numbers from 0 to 9 - appearing in different percentages. These are, respectively: 2%, 10%, 4%, 14%, 2%,18%, 6%, 16%, 10%, 18%. Recall that to obtain probabilities, you need to divide the percentage values by 100. Thus, we get 0.02; 0.1, etc. Let us present an example of solving the problem for the variance of a random variable and the mathematical expectation.

We calculate the arithmetic mean using the formula that we remember from elementary school: 50/10 = 5.

Now let’s convert the probabilities into the number of outcomes “in pieces” to make it easier to count. We get 1, 5, 2, 7, 1, 9, 3, 8, 5 and 9. From each value obtained, we subtract the arithmetic mean, after which we square each of the results obtained. See how to do this using the first element as an example: 1 - 5 = (-4). Next: (-4) * (-4) = 16. For other values, do these operations yourself. If you did everything correctly, then after adding them all up you will get 90.

Let's continue calculating the variance and expected value by dividing 90 by N. Why do we choose N rather than N-1? Correct, because the number of experiments performed exceeds 30. So: 90/10 = 9. We got the variance. If you get a different number, don't despair. Most likely, you made a simple mistake in the calculations. Double-check what you wrote, and everything will probably fall into place.

Finally, remember the formula for mathematical expectation. We will not give all the calculations, we will only write an answer that you can check with after completing all the required procedures. The expected value will be 5.48. Let us only recall how to carry out operations, using the first elements as an example: 0*0.02 + 1*0.1... and so on. As you can see, we simply multiply the outcome value by its probability.

Deviation

Another concept closely related to dispersion and mathematical expectation is standard deviation. It is designated either with Latin letters sd, or Greek lowercase "sigma". This concept shows how much on average the values deviate from the central feature. To find its value, you need to calculate Square root from dispersion.

If you plot a normal distribution graph and want to see the squared deviation directly on it, this can be done in several stages. Take half of the image to the left or right of the mode (central value), draw a perpendicular to the horizontal axis so that the areas of the resulting figures are equal. The size of the segment between the middle of the distribution and the resulting projection onto the horizontal axis will represent the standard deviation.

Software

As can be seen from the descriptions of the formulas and the examples presented, calculating variance and mathematical expectation is not the simplest procedure from an arithmetic point of view. In order not to waste time, it makes sense to use the program used in higher education institutions - it is called “R”. It has functions that allow you to calculate values for many concepts from statistics and probability theory.

For example, you specify a vector of values. This is done as follows: vector<-c(1,5,2…). Теперь, когда вам потребуется посчитать какие-либо значения для этого вектора, вы пишете функцию и задаете его в качестве аргумента. Для нахождения дисперсии вам нужно будет использовать функцию var. Пример её использования: var(vector). Далее вы просто нажимаете «ввод» и получаете результат.

Finally

Dispersion and mathematical expectation are without which it is difficult to calculate anything in the future. In the main course of lectures at universities, they are discussed already in the first months of studying the subject. It is precisely because of the lack of understanding of these simple concepts and the inability to calculate them that many students immediately begin to fall behind in the program and later receive bad grades at the end of the session, which deprives them of a scholarship.

Practice for at least one week, half an hour a day, solving tasks similar to those presented in this article. Then, on any test in probability theory, you will be able to cope with the examples without extraneous tips and cheat sheets.

– the number of boys among 10 newborns.

It is absolutely clear that this number is not known in advance, and the next ten children born may include:

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

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

– long jump distance (in some units).

Even a master of sports cannot predict it :)

However, your hypotheses?

2) Continuous random variable – accepts All numerical values from some finite or infinite interval.

Note : the abbreviations DSV and NSV are popular in educational literature

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

Distribution law of a discrete random variable

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

The term is used quite often row distribution, but in some situations it sounds ambiguous, and so I will stick 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 condensed:

So, for example, the law of probability distribution of points rolled 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 winning distribution law:

...you've probably dreamed of such tasks for a long time :) I'll tell you a secret - me too. Especially after I finished working on field theory.

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

Exposing the “partisan”:

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

Control: that’s what we needed to make sure of.

Answer:

It is not uncommon when you need to draw up a distribution law yourself. For this they use classical definition of probability, multiplication/addition theorems for event probabilities and other chips tervera:

Example 2

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

Solution: as you noticed, the values of a random variable are usually placed in in ascending order. Therefore, we start with the smallest winnings, namely rubles.

There are 50 such tickets in total - 12 = 38, and according to classical definition:
– the probability that a randomly drawn ticket will be a loser.

In other cases everything is simple. The probability of winning rubles is:

Check: – and this is a particularly pleasant moment of such tasks!

Answer: the desired law of distribution of winnings:

The following task is for you to solve on your own:

Example 3

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

...I knew that you missed him :) Let's remember multiplication and addition theorems. The solution and answer are at the end of the lesson.

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

Expectation of a discrete random variable

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

or collapsed:

Let us calculate, for example, the mathematical expectation of a random variable - the number of points rolled on a die:

Now let's remember our hypothetical game:

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

Thus, the mathematical expectation of this game losing.

Don't trust your impressions - trust the numbers!

Yes, here you can win 10 or even 20-30 times in a row, but in the long run, inevitable ruin awaits us. And I wouldn't advise you to play such games :) Well, maybe only for fun.

From all of the above it follows that the mathematical expectation is no longer a RANDOM value.

Creative task for independent research:

Example 4

Mr. X plays European roulette using the following system: he constantly bets 100 rubles on “red”. Draw up a law of distribution of a random variable - its winnings. Calculate the mathematical expectation of winnings and round it to the nearest kopeck. How many average Does the player lose for every hundred he bet?

Reference : European roulette contains 18 red, 18 black and 1 green sector (“zero”). If a “red” appears, the player is paid double the 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 or tables, because it has been established for certain that the player’s mathematical expectation will be exactly the same. The only thing that changes from system to system is

01.02.2018

Expected value. Just something complicated. Basics of trading.

When placing bets of any type, there is always a certain probability of making a profit and a 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 dwell in detail on these two aspects of trading.

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

The idea is that a positive expected value leads to positive (profit-enhancing) trading, while a zero or negative expected value means no trading at all.

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

Roulette- (The dealer launches the ball in the opposite direction of 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 on 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 negative mathematical expectation. It's all because of the zero field, which is neither black nor red.

Because (in general) if the bet change is not applied, the player loses $1 for every 37 spins of the wheel (at a bet of $1 at a time), resulting in a linear loss of -2.7%, which increases as the number of bets increases (average).

Of course, over an interval of, for example, 1000 games, a player may experience a series of victories, and a person may begin to mistakenly believe that he can earn money by beating the casino, as well as a series of defeats. A series of victories in this case can increase the player’s capital by a greater value than he initially had, 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 ends up with less or more money, we will call this difference between the current capital variance. You can make money playing roulette only within the framework of variance. If the player continues to follow this strategy, ultimately the person will be left without money, and the casino will make money.

The second example is the famous binary options. They let you place a bet, if the outcome is successful, you take as much as 90 percent of your bet on top, and if it’s unsuccessful, you lose all 100. And then the BO owners just have to wait, the market and the negative checkmate expectation will do their job. And time dispersion will give hope to the binary options trader that it is possible to make money on 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. (And if the situation with the second one is quite controversial, then the risk needs to be controlled very clearly.)

To understand in which direction your strategy is leading you, you need to maintain statistics. A trader should know:

The number of your trades. The greater the number of trades for a given strategy, the more accurate the mathematical expectation will be
Frequency of successful entries. (Probability) (R)
Your profit for each positive transaction.
Bias (win rate) (B)
Average size of your bet (stop order) (S)

Mathematical expectation (E) = B * R – (1 – B) = B * (1 + R) –1

To roughly find out your total earnings or losses on your account (EE), for example, over 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.

profit - 100 dollars.

Number of transactions 30

The 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.

Let now the profit be 150. Then the checkmate expectation will be negative at a ratio of 16%/84%. Or lower.

Conclusion.

What to do about it? Start keeping statistics if you haven't already. Check your trades, determine your checkmate expectation. Find what can be improved (number of correct entries, gaining profit, cutting losses)

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