Mathematical expectation M x is equal. Random variables
Mathematical expectation is the distribution of random variance probabilities
Mathematical expectation, definition, mathematical expectation of discrete and continuous random variables, selective, conditional matchmaking, calculation, properties, tasks, assessment of matchmakers, dispersion, distribution function, formula, calculation examples
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Mathematical expectation is a definition
One of the most important concepts in mathematical statistics and the theory of probabilities, which characterizes the distribution of values \u200b\u200bor probabilities of a random variable. Usually expressed as the weighted average value of all possible random variance parameters. Widely used when conducting technical analysis, research numerical rows, studying continuous and long-lasting processes. It is important in assessing risks, predicting price indicators in trade in financial markets, is used in the development of strategies and methods of game tactics in the theory of gambling.
Mathematical expectation isthe average value of a random variable, the distribution of probabilities of a random variable is considered in the theory of probability.
Mathematical expectation isthe measure of the average value of the random variable in the theory of probability. Mathematical expectation of a random variable x. denotes M (X).
Mathematical expectation is
Mathematical expectation is In the theory of probability, the weighted average value of all possible values \u200b\u200bthat this random value can take.
Mathematical expectation isthe amount of works of all possible values \u200b\u200bof random variance on the likelihood of these values.
Mathematical expectation is The average benefit from one or another solution, provided that such a solution can be considered within the framework of the theory of large numbers and a long distance.
Mathematical expectation isin the theory of gambling, the amount of winnings, which can earn or lose a player, on average, at each rate. In the language of gambling players, this is sometimes called the "advantage of the player" (if it is positive for the player) or the "advantage of the casino" (if it is negative for the player).
Mathematical expectation is The percentage of profit on winnings multiplied by the average profit, minus the probability of a loss multiplied by the average loss.
Mathematical expectation of random variable in mathematical theory
One of the important numerical characteristics of a random variable is expected value. We introduce the concept of a system of random variables. Consider a combination of random variables that are the results of the same random experiment. If - one of the possible system values, the event corresponds to a certain probability that satisfy the axioms of Kolmogorov. The function defined in any possible values \u200b\u200bof random variables is called the joint distribution law. This feature allows you to calculate the likelihood of any events from. In particular, the joint law of the distribution of random variables and that take values \u200b\u200bfrom the set and are given by the probabilities.
The term "mathematical expectation" was introduced by Pierre Simon Marquis de Laplas (1795) and occurred from the concept of the "expected value of the winnings", which first appeared in the 17th century in the theory of gambling in the works of Blaise Pascal and Christian Guygens. However, the first complete theoretical understanding and evaluation of this concept is given by Paphing Lvivich Chebyshev (the middle of the 19th century).
The law of the distribution of random numeric values \u200b\u200b(distribution function and distribution range or probability density) fully describe the behavior of a random value. But in a number of tasks it is enough to know some numerical characteristics of the value under study (for example, its average value and possible deviation from it) to respond to the assigned question. The main numeric characteristics of random variables are mathematical expectation, dispersion, mod and median.
The mathematical expectation of the discrete random variable is the amount of products of its possible values \u200b\u200bto the probability corresponding to them. Sometimes mathematical expectation is called a weighted average, as it is approximately equal to the average arithmetic observed values \u200b\u200bof the random variable with a large number of experiments. From the determination of mathematical expectation it follows that its value is not less than the smallest possible value of the random variable and not more than the largest. The mathematical expectation of a random variable is the non-random (constant) value.
Mathematical expectation has a simple physical meaning: if there is a single mass on a straight line, placing some mass (for discrete distribution), or "folding" it with a certain density (for absolutely continuous distribution), the point corresponding to the mathematical expectation will be the coordinate "The center of gravity" is straight.
The average value of random variance is a number that seems to be its "representative" and replacing it with roughly approximate calculations. When we say: "The average lamp operation is 100 o'clock" or "The average point of contact is shifted relative to the target to 2 m to the right," we indicate this a certain numerical characteristic of a random variable describing its location on the numerical axis, i.e. "The characteristic of the situation."
From the characteristics of the position in the theory of probability, the mathematical expectation of a random variable plays, which is sometimes called simply the average value of a random variable.
Consider a random amount H.having possible values x1, x2, ..., xn With probabilities p1, P2, ..., PN. We need to characterize some number the position of the values \u200b\u200bof the random variable on the abscissa axis, taking into account the fact that these values \u200b\u200bhave different probabilities. For this purpose, it is natural to use the so-called "average weighted" from the values xI, Moreover, each XI value with averaging must be taken into account with the "weight" proportional to the probability of this value. Thus, we calculate the average random variable X.we denote M | x |:
This is a secondary value and is called the mathematical expectation of a random variable. Thus, we introduced in consideration one of the most important concepts of probability theory is the concept of mathematical expectation. The mathematical expectation of a random variety is called the amount of products of all possible values \u200b\u200bof random variance on the likelihood of these values.
H. associated with a peculiar dependence with the average arithmetic observed values \u200b\u200bof a random variable with a large number of experiments. This dependence of the same type as the relationship between the frequency and probability, namely, with a large number of experiments, the average arithmetic observed values \u200b\u200bof the random variable approaching (converges in probability) to its mathematical expectation. From the presence of communication between the frequency and probability can be derived as a consequence of the presence of a similar connection between the average arithmetic and mathematical expectation. Indeed, consider a random amount H.characterized by a number of distribution:
Let it be produced N. independent experiments, in each of which the amount X.takes a certain value. Suppose that value x1.appeared m1.times, meaning x2appeared m2.once, general value xImi appeared once. Calculate the average arithmetic observed values \u200b\u200bof the amount of X, which, in contrast to the mathematical expectation M | x |we denote M * | x \u200b\u200b|:
With an increase in the number of experiments N.frequency pIwill be approached (converge in probability) to the appropriate probabilities. Therefore, the average arithmetic observed values \u200b\u200bof the random variable M | x | With an increase in the number of experiments, it will approach (converge in probability) to its mathematical expectation. The above relationship between the average arithmetic and mathematical expectation is the content of one of the forms of the law of large numbers.
We already know that all forms of the law of large numbers state the fact of sustainability of some medium with a large number of experiments. Here we are talking about the stability of the average arithmetic from a number of observations of the same value. With a small number of experiments, the arithmetic average of their results randomly; With a sufficient increase in the number of experiments, it becomes "almost no accident" and, stabilizing, is approaching a constant value - mathematical expectation.
The property of the sustainability of the average with a large number of experiments is easy to check experimentally. For example, weighing any body in the laboratory on the exact scales, we obtain the new value as a result of weighing each time; To reduce the observation error, we weigh the body several times and use the average arithmetic values. It is easy to make sure that with the further increase in the number of experiments (weighing), the average arithmetic reacts to this increase is less and less and with a sufficiently large number of experiments almost ceases to change.
It should be noted that the most important characteristic of the position of the random variable is a mathematical expectation - there is not for all random variables. You can create examples of such random variables for which the mathematical expectation does not exist, since the corresponding amount or integral is diverted. However, such cases are not significant for practice. Usually, the random variables with which we are dealing with a limited area of \u200b\u200bpossible values \u200b\u200band, of course, have mathematical expectations.
In addition to the most important of the characteristics of the position of a random variable - mathematical expectation, in practice sometimes other characteristics of the position, in particular, fashion and median of a random variable are also applied.
The fashion of the random variable is called its most likely value. The term "most likely value", strictly speaking, applies only to interrupted values; For a continuous magnitude of the fashion, the value in which the probability density is maximal. Figures show the Fashion, respectively, for intermittent and continuous random variables.
If the distribution polygon (distribution curve) has more than one maximum, the distribution is called "polymodal".
Sometimes there are distributions that possess in the middle not a maximum, and minima. Such distributions are called "antimodal".
In general, fashion and mathematical expectation of random variance do not coincide. In the particular case, when the distribution is symmetric and modal (that is, it has a fashion) and there is a mathematical expectation, it coincides with the fashion and the distribution symmetry center.
It is often used another position characteristic - the so-called median of a random variety. This characteristic is usually used only for continuous random variables, although it is possible to define it for intermittently values. The geometrically median is the abscissa of the point in which the area, limited distribution curve, is divided into half.
In the case of a symmetric modal distribution, the median coincides with mathematical expectation and fashion.
The mathematical expectation is an average value, random variable - the numerical characteristic of the probability distribution of random variable. The most common mathematical expectation of a random variable X (W) Determined as the Lebek integral in relation to probability Rin the initial probabilistic space:
Mathematical expectation can be calculated and as a Lebesgue integral from h.by the distribution of probabilities rHvalues X.:
Naturally, it is possible to determine the concept of a random variable with an infinite mathematical expectation. A typical example is the time of return in some random wanders.
With the help of mathematical expectations, many numeric and functional characteristics of the distribution are determined (as a mathematical waiting for the corresponding functions from a random variable), for example, producing a function, a characteristic function, moments of any order, in particular dispersion, covariance.
The mathematical wait is the characteristic of the location of random values \u200b\u200b(the average value of its distribution). In this capacity, the mathematical exercise serves as a "typical" distribution parameter and its role is similar to the role of a static moment - the coordinates of the center of gravity of the mass distribution - in mechanics. From the other characteristics of the location, by which the distribution is described in general terms, the median, mod, the mathematical expectation is of the greatest value that it and the scattering characteristic corresponding to it is dispersion - 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 inequality) and the enhanced law of large numbers.
Mathematical expectation of a discrete random variable
Let there be some random value that can take one of several numeric values \u200b\u200b(for example, the number of points when throwing a bone can be 1, 2, 3, 4, 5 or 6). Often, the question arises in practice for such a magnitude: what value does it take "on average" with a large number of tests? What will be our average income (or loss) from each of the risky operations?
Say, there is some kind of lottery. We want to understand, it is advantageous or not to participate in it (or even participate repeatedly, regularly). Suppose a winning every fourth ticket, the prize will be 300 rubles, and the price of any ticket is 100 rubles. With an infinitely large number of participation, it turns out that. In three quarters, we will lose, every three losses will cost 300 rubles. In each fourth case, we will win 200 rubles. (A prize minus cost), that is, in four participation we are on average we lose 100 rubles, for one - on average 25 rubles. Total on average the rates of our ruin will be 25 rubles / ticket.
We throw a playing bone. If it is not a scaling (without shifting the center of gravity, etc.), how much will we all have glasses at a time? Since each variant is equally intended, we take stupidly arithmetic and we get 3.5. Since it is average, there is no need to indignant that 3.5 points no specific throw will not give - well, there is no place for this cube with such a number!
Now we generalize our examples:
Turn to the just shown picture. On the left distribution plate of the random variable. X value can take one of the n possible values \u200b\u200b(are given in the top line). No other values \u200b\u200bmay not be. Under each possible value, its probability is signed below. The right is a formula where M (x) is called mathematical expectation. The meaning of this magnitude is that with a large number of tests (with a large sample), the average value will strive for this very mathematical expectation.
Let's return again to the same playful Cuba. The mathematical expectation of the amount of points when throwing is 3.5 (count themselves according to the formula, if you do not believe). Let's say you threw it a couple of times. 4 and 6 fell out. On average, it turned out 5, that is, away from 3.5. They threw another time, it fell 3, that is, on average (4 + 6 + 3) / 3 \u003d 4,3333 ... somehow far from the mathematical expectation. Now spend a crazy experiment - throw a cube 1000 times! And if on average and there will be no exactly 3.5, it will be close to that.
We calculate the mathematical expectation for the above-described lottery. The sign will look like this:
Then the mathematical expectation will be as we set up above.:
Another thing is that the same "on the fingers", without a formula, it would be difficult if there were more options. Well, let's say, there would be 75% of losing tickets, 20% of winning tickets and 5% of particularly advantageous.
Now some properties of mathematical expectation.
Prove it just:
A permanent multiplier is allowed to be made for a sign of mathematical expectation, that is:
This is a special case of the properties of the limit of mathematical expectation.
Other consequence of the linearity of 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:
It is also easy to prove) XY. itself is a random amount, with the initial values \u200b\u200bcould take n.and m.values, respectively, then XY.can take NM values. The probability of each of the values \u200b\u200bis calculated based on the fact that the probabilities of independent events are variable. In the end, we get this:
Mathematical expectation of a continuous random variable
In continuous random variables, there is such a characteristic as the distribution density (probability density). She, in essence, characterizes the situation that some values \u200b\u200bfrom a variety of valid numbers random value takes more often, some less often. For example, consider this schedule:
Here X.- Actually random variable, f (x)- Distribution density. Judging by this schedule, with the experiments value X.it will often be a number close to zero. Chances to exceed 3 or to be less -3 rather, pure theoretical.
Let, for example, there is a uniform distribution:
This fully corresponds to an intuitive understanding. For example, if we get with a uniform distribution a lot of random valid numbers, each of the segment |0; 1| , The arithmetic average should be about 0.5.
The properties of mathematical expectation are linearity, etc., applicable to discrete random variables, applicable here.
The relationship of mathematical expectation with other statistical indicators
In statistical analysis, along with a mathematical expectation, there is a system of interdependent indicators reflecting the homogeneity of phenomena and the stability of processes. Often, the indicators of the variation do not have an independent meaning and are used to further analyze the data. The exception is the variation coefficient, 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 Dispersionwhich is the most close and directly related to the mathematical expectation. This parameter is actively used in other types of statistical analysis (testing hypotheses, analysis of causal relationships, etc.). Like the average linear deviation, the dispersion also reflects the measure of data scatter around the average value.
Language of signs is useful to translate into the language of words. It turns out that the dispersion is the middle square of deviations. That is, at first the average value is calculated, then the difference is taken between each source and average value, it is erected into a square, it is also divided into the number of values \u200b\u200bin this set. The difference between the individual value and the average reflects the deviation measure. The square is built to ensure that all deviations have become exclusively positive numbers And to avoid interconnection of positive and negative deviations when summarizing them. Then, having squares of deviations, we simply calculate the average arithmetic. Middle - Square - deviations. Deviations are elevated in a square, and the average is considered. The impact of the magical word "dispersion" lies in three words.
However, in its pure form, such as the average arithmetic, or index, the dispersion is not used. It is rather auxiliary and intermediate indicator, which is used for other types of statistical analysis. She even has no normal units. Judging by the formula, this is the square of the unit of measurement of the source data.
Let we measure the random variable N.once, for example, we measure the wind speed and we want to find the average value. How is the mean value with the distribution function?
Or we will throw a playing cube a large number of times. The number of points that falls on the cube with each throw is a random value and can take any natural values \u200b\u200bfrom 1 to 6. The average arithmetic pulmoned points counted for all the casts of the cube is also a random variable, but with large N.it seeks to quite specific number - Mathematical expectation MX.. In this case, MX \u003d 3.5.
How did this value come out? Let B N.tests n1.once fell 1 point, n2.once - 2 points and so on. Then the number of outcomes in which one point fell:
Similarly, for outcomes, when 2, 3, 4, 5 and 6 points fell.
Suppose now that we know the law of the distribution of the random value x, that is, we know that the random value of X can take values \u200b\u200bx1, x2, ..., xk with probabilities P1, P2, ..., PK.
Mathematical expectation MX random variance x is:
Mathematical expectation is not always a reasonable assessment of some random variety. So, to estimate the average wage, it is more reasonable to use the concept of median, that is, such a value that the number of people receiving less than the median, salary and large, coincide.
The probability P1 is that the random variable will be less than x1 / 2, and the probability of P2 is that the random value of X is greater than x1 / 2, the same and equal to 1/2. The median is defined uniquely not for all distributions.
Standard or standard deviation In statistics, the degree of deviation of observation data or sets from the mean value is called. Denoted by letters s or s. A small standard deviation indicates that the data is grouped around the average value, and significant - that the initial data is located far from it. Standard deviation is equal square root Values \u200b\u200bcalled dispersion. It is the average number of the sum of the initial data differences deviating from the average value. The standard deviation of the random variable is called the root square from the dispersion:
Example. Under the conditions of testing when shooting a target, calculate the dispersion and the riconductic deviation of the random variable:
Variation- oscillating, variability of the sign of a sign in units of aggregate. Separate numerical values \u200b\u200bof the feature found in the aggregate is called variants. The insufficiency of the average value for the complete characteristics of the aggregate makes supplement the average values \u200b\u200bof the indicators that allow us to estimate the typicity of these mean by measuring the varying (variations) of the studied sign. The variation coefficient is calculated by the formula:
Variation variation (R) represents the difference between the maximum and minimum values \u200b\u200bof the feature in the common totality. This indicator gives the most general view On the sections of the studied sign, as it shows the difference only between the limit values \u200b\u200bof the options. The dependence on the extreme values \u200b\u200bof the attribute gives the scope of the variation is unstable, random character.
Medium linear deviationit is the arithmetic average of the absolute (module) deviations of all values \u200b\u200bof the analyzed aggregate from their average size:
Mathematical expectation in gambling theory
Mathematical expectation isthe average amount of money that gambling a player can win or lose at this rate. This is a very significant concept for a player, because it is fundamental to assess the majority of play situations. The mathematical expectation is also an optimal tool for analyzing the main card layouts and play situations.
Suppose you play with a friend in a coin, every time making a bet ride for $ 1, regardless of what will fall. The rush - you won, the eagle - lost. The chances of what the rush will fall one to one, and you bet $ 1 to $ 1. Thus, the mathematical expectation is zero, because From the point of view of mathematics, you can't know you will behave or play after two shots or after 200.
Your watch win is zero. The clock gain is the amount of money you expect to win in an hour. You can throw a coin 500 times within an hour, but you will not win and do not lose, because Your chances are not positive, nor negative. If you look, from the point of view of a serious player such a system of bets. But this is simply a time loss.
But suppose someone wants to put $ 2 against your $ 1 in the same game. Then you immediately have a positive matchmaker in 50 cents from each bet. Why 50 cents? On average, one bet you won, the second losing. Put the first dollar - and lose $ 1, put the second - win $ 2. You made a $ 1 bet twice and go ahead for $ 1. Thus, each of your one-dollar bets gave you 50 cents.
If in one hour the coin will fall 500 times, your watch winnings will already be $ 250, because On average, you lost one dollar 250 times and won two dollars 250 times. $ 500 minus $ 250 is $ 250, which is the total win. Please note that the matchmaker, which is the amount that you have won on the same rate, is equal to 50 cents. You won $ 250, making a bet on the dollar 500 times, which is equal to 50 cents from the bet.
Mathematical expectation has nothing to do with short-term results. Your opponent who decided to put $ 2 against you could beat you on the first ten throws in a row, but you, possessing the advantage of bets 2 to 1, with other things being equal, you earn 50 cents from each rate of $ 1 in any circumstances. There is no difference, you win either lose one bet or several rates, but only if you have enough cash to quietly compensate the costs. If you continue to install the same time, for a long period of time, your winnings will suit the sum of the matchmakers in individual throws.
Each time, betting a bet with the best outcome (a bet that can be beneficial on a long distance), when the chances of your favor, you will definitely win something on it, and it does not matter to lose it or not in this hand. And on the contrary, if you made a bet with a worst outcome (a bet that is unprofitable on a long distance), when the chances are not in your favor, you lose something no matter what you won or lost in this hand.
You bet with the best outcome if you have a positive match, and it is positive if the chances are on your side. Making a bet with a worst outcome, you have a negative matchmaker that happens when chances against you. Serious players make bets only with the best outcome, at worst - they will graze. What does the chances mean to your favor? You can eventually win more than you bring real chances. The real chances of what the rush will fall 1 to 1, but you have 2 to 1 due to the ratio of rates. In this case, the chances of your favor. You exactly get the best outcome with a positive expectation of 50 cents per bet.
Here is a more complex example of mathematical expectation. The buddy writes numbers from one to five and bets $ 5 against your $ 1 to the fact that you do not define the specified number. Do you agree on such a bet? What is the matchmaker here?
On average, four times you will be mistaken. Based on this, the chances against the fact that you are guessing the figure will be 4 to 1. The chances for the fact that with one attempt you lose the dollar. Nevertheless, you win 5 to 1, if possible to lose 4 to 1. Therefore, the chances of your favor, you can take bets and hope for the best outcome. If you make such a bet five times, on average you will lose four times $ 1 and win $ 5 once. Based on this, for all five attempts you earn $ 1 with a positive mathematical expectation of 20 cents per bet.
A player who is going to win more than puts, as in the example above, catches chances. And on the contrary, he ruins the chances when he assumes to win less than puts. A bet player may have either a positive or negative match-term, which depends on whether he catches or ruins the chances.
If you put $ 50 in order to win $ 10 at the probability of winning 4 to 1, then you will receive a negative match-term $ 2, because On average, you will win four times at $ 10 and will play $ 50 once, which shows that the loss in one bet will be $ 10. But if you put $ 30 in order to win $ 10, with the same chances of winning 4 to 1, then in this case you have a positive wait of $ 2, because You again win four times at $ 10 and plays $ 30 once, which will make a profit of $ 10. These examples show that the first bet is bad, and the second is good.
The mathematical expectation is the center of any gaming situation. When a bookmaker encourages football fans to raise $ 11 to win $ 10, it has a positive matchmaker from every $ 10 in the amount of 50 cents. If the casino pays equal money from the passage line in a fastener, the positive wait of the casino will be approximately $ 1.40 every $ 100, because This game is built so that everyone who put on this line loses on average 50.7% and wins 49.3% of total time. Undoubtedly, it is this kind of minimal positive matchmakers and brings colossal profits to casino owners around the world. As the owner of the VEGAS World Casino's owner, Bob Stupak, "one thousandth percent of the negative probability on a sufficiently long distance will be dispersed rich man in the world".
Mathematical expectation when playing poker
Poker game is the most indicative and visual example from the point of view of using the theory and properties of mathematical expectation.
The mathematical expectation (English Expected Value) in poker is the average benefit from one or another solution, provided that such a decision can be considered within the framework of the theory of large numbers and a long distant. A successful poker game is to always take moves only with a positive mathematical expectation.
The mathematical meaning of the mathematical expectation when playing poker is that we are often encountered with random values \u200b\u200bwhen making a decision (we do not know which cards in the hands of the opponent, which cards will come on subsequent trading circles). We must consider each of the solutions from the point of view of the theory of large numbers, which states that with a sufficiently large sample, the average value of a random variable will strive for its mathematical expectation.
Among private formulas for calculating mathematical expectations, the most applied in poker is the following:
When playing poker mathematical expectation, you can count on both for bets and collov. In the first case, Fold Equiti should be taken into account, in the second - the bank's own chances of the bank. When evaluating the mathematical expectation of a turn, it should be remembered that Fold always has zero matching. Thus, the discharge of maps will always be a more profitable solution than any negative move.
Waiting tells you about what you can expect (profit or loss) for every dollar at your risk. Casino earn money because the mathematical expectation from all the games that are practiced in them, in favor of the casino. With a sufficiently long series of the game, you can expect the client to lose its money because the "probability" in favor of the casino. However, professional players in the casino limit their games with short intervals, thereby increasing the likelihood in their favor. The same applies to investment. If your wait is positive, you can earn more money, Making a lot of transactions in a short period of time. Waiting This is your percentage of profit on winning, multiplied by the average profit, minus your probability is a loss multiplied by an average loss.
Poker can also be considered from the point of view of mathematical expectation. You may assume that a certain course is beneficial, but in some cases it may be far from the best, because it is more profitable another move. Suppose you have collected a full-house in a five-recurrent poker with an exchange. Your rival bets. You know that if you raise the bet, he will answer. Therefore, the increase looks like better tactics. But if you still raise the bid, the remaining two players will definitely drop the cards. But if you equalize the bid, you will be completely sure that the two other players will arrive after you. When raising rates, you get one unit, and simply equalizing - two. Thus, the equalization gives you a higher positive mathematical expectation, and will be the best tactics.
A mathematical expectation can also give the concept of which in poker tactics is less profitable, and what more. For example, playing on a certain hand, you believe that your losses on average will make up 75 cents, including ante, then such a hand should be played, because It is better than reset when Ante is $ 1.
Another important reason for understanding the essence of the mathematical expectation is that it gives you a feeling of calm, regardless of whether you have won the bid or not: if you have made a good bet or saved you, you will know that you have earned or saved a certain amount of money that you The player was weaker could not save. It is much more difficult to reset the cards if you are upset that the opponent in the exchange collected a stronger combination. With all this, the money you saved, without playing, instead of putting, adding to your win per night or for the month.
Just remember that if you change your hands, your opponent would answer you, and as you will see in the article "Fundamental Poker Theorem" is just one of your advantages. You must rejoice when it happens. You can even learn to enjoy the lost distribution, because you know that other players would lose much more.
As mentioned in the example with a coin game at the beginning, the hourly profit factor is interrelated with mathematical expectation, and this concept Especially important for professional players. When you are going to play poker, you must mentally estimate how much you can win in the hour of the game. In most cases, you will need to be based on your intuition and experience, but you can also use some mathematical calculations. For example, you play a lobol with an exchange, and watch that the three participants make rates on $ 10, and then change two cards, which is very bad tactics, you can count for yourself that every time they put $ 10, they lose About $ 2. Each of them makes it eight times a hour, which means that all three are losing at an hour about $ 48. You are one of the remaining four players who are approximately equal, accordingly, these four players (and you among them) must divide $ 48, and each profit will be $ 12 per hour. Your clock coefficient in this case is simply equal to your share from the amount of money played in three bad players per hour.
For a large period of time, the total winning player is the amount of its mathematical expectations in separate distribution. The more you play with a positive expectation, the more win, and vice versa, the more distributions with a negative expectation you will play, the more you lose. As a result, the game should be preferred, which will be able to maximize your positive wait or will not be negative, so that you can raise your watch wisness to the maximum.
Positive mathematical expectation in the gaming strategy
If you know how to count the cards, you may have an advantage over the casino, if they do not notice it and do not throw you out. Casino adore drunk players and do not tolerate considered cards. The advantage will allow you over time to win more than once than to lose. Good capital management when using mathematical expectation calculations can help extract more profits from your advantage and reduce losses. Without the advantage you better give money for charity. In the game on the stock exchange, the advantage gives a game system that creates a big profit than the loss, price difference and commission. No capital management will save the bad game system.
Positive waiting is determined by a value exceeding zero. The larger this number, the stronger the statistical wait. If the value is less than zero, the mathematical expectation will also be negative. The larger the negative module, the worse the situation. If the result is zero, then the expectation is abrupt. You can win only when you have a positive mathematical expectation, a reasonable game system. The intuition game leads to a catastrophe.
Mathematical Waiting and Exchange Trade
The mathematical expectation is a fairly popular and popular statistical indicator in the implementation of exchange trading in the financial markets. First of all, this parameter is used to analyze the success of trading. It is not difficult to guess that the more this value, the more reason to consider the trading successful trade. Of course, the analysis of the work of the trader cannot only be made using this parameter. However, the calculated value in aggregate with other ways to assess the quality of work can significantly increase the accuracy of the analysis.
The mathematical expectation is often calculated in the services of monitoring accounts, which allows you to quickly evaluate the work performed on the deposit. As exceptions, it is possible to bring strategies in which the "reinforcement" of unprofitable transactions is used. The trader can accompany a luck for some time, and therefore in his work may not be losses in general. In this case, it will not be possible to navigate only in the battalion, because the risks used in the work will not be taken into account.
In market trade, mathematical expectation is most often used in predicting the profitability of any trading strategy or when predicting the trader's income based on the statistical data of its previous trading.
Regarding capital management it is very important to understand that when making transactions with a negative expectation there is no money management scheme that can definitely bring high profits. If you continue to play on the stock exchange in these conditions, then regardless of the method of money management you will lose your entire account, no matter how big it is at the beginning.
This axiom is true not only for playing or deals with a negative expectation, it is also true for playing with equal chances. Therefore, the only case when you have a chance to benefit in the long run, is the conclusion of transactions with a positive mathematical expectation.
The difference between negative expectations and positive expectations is the difference between life and death. It does not matter how positive or as far as negative expectation; It is only important that it is positive or negative. Therefore, before consideration of capital management issues, you must find the game with a positive expectation.
If you have no such game, then no money management in the world will save you. On the other hand, if you have a positive wait, then you can, through the proper money management, turn it into the function of exponential growth. It does not matter how little it is a positive wait! In other words, it does not matter how profitable the trading system is based on a single contract. If you have a system that wins 10 dollars to a contract in one transaction (after the deduction of commission and slippage), you can use capital management methods in such a way as to make it more profitable than the system that shows an average profit of $ 1000 for the transaction (after deductions for commission and slippage).
It does not matter how profitable the system was, and how definitely it can be said that the system will show at least minimal profits in the future. Therefore, the most important preparation that a trader can make is to make sure that the system will show a positive mathematical expectation in the future.
In order to have a positive mathematical expectation in the future, it is very important not to limit the degrees of freedom of your system. This is achieved not only by abolishing or decreasing the number of parameters to be optimized, but also by reducing the system as much as possible. Each parameter you add, each rule that you make, every smallest change that you do in the system reduces the number of degrees of freedom. Ideally, you need to build a fairly primitive and simple system that will constantly bring a small profit by almost any market. And again it is important that you understand, it doesn't matter how profitable the system is until it is profitable. Money that you earn in trade will be earned by effective money management.
The trading system is just a tool that gives you a positive mathematical expectation so that you can use the money management. Systems that work (show at least minimum profits) only in one or several markets or have different rules or parameters for different markets, most likely will not work in real time long enough. The problem of most technically oriented traders is that they spend too much time and efforts to optimize various rules and values \u200b\u200bof the parameters of the trading system. This gives completely opposite results. Instead of spending strength and computer time to increase the profits of the trading system, send the energy to increase the level of reliability of minimal profits.
Knowing that capital management is just a numerical game that requires the use of positive expectations, the trader may stop the search for "sacred grail" of exchange trading. Instead, he can do the check of his trading method, find out how logically justified this method, whether he gives pollen expectations. The correct methods of capital management, applied in relation to any, even very mediocre trade methods, will make them all the rest.
To any trader to succeed in his work, it is necessary to solve the three most important tasks :. Ensure that the number of successful transactions exceeds the inevitable errors and miscalculations; Customize your trading system so that the possibility of earning is as often as possible; Achieve the stability of the positive result of their operations.
And here we, working traders, a good help can have mathematical expectation. This term in the theory of probability is one of the key. With it, it is possible to give averaged assessment to some random meaning. The mathematical expectation of random variance is similar to the center of gravity, if you imagine all the possible probabilities with dots with different mass.
With regard to the trading strategy, mathematical expectation of profit (or loss) is most often used to evaluate its effectiveness. This parameter is determined as the amount of the works of the specified levels of profit and loss and the probabilities of their appearance. For example, the developed trade strategy assumes that 37% of all operations will cause profits, and the remaining part is 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 dollars. Let's calculate the mathematical expectation of trade on such a system:
What does this number mean? It suggests that following the rules of this system, on average we will receive 1.708 dollars from each closed transaction. Since the resulting evaluation estimate is greater than zero, then such a system can be used for real work. If, as a result of the calculation, the mathematical expectation will be negative, it is already talking about an average damage and such trade will lead to a ruin.
The amount of profit per one transaction can also be expressed and the relative value in the form of%. For example:
- income percentage of 1 transaction - 5%;
- the percentage of successful trading operations - 62%;
- percentage of loss per 1 transaction - 3%;
- the percentage of unsuccessful transactions - 38%;
That is, the average transaction will bring 1.96%.
You can develop a system that despite the prevalence of unprofitable transactions will give a positive result, since its MO\u003e 0.
However, one expectation is small. It is difficult to earn if the system gives very little trading signals. In this case, its yield will be comparable to a bank percentage. Let each operation give an average of only 0.5 dollars, but what if the system assumes 1000 operations per year? It will be a very serious amount for a relatively small time. It logically implies that another distinguishing sign of a good trading system can be considered a short period of holding positions.
Sources and links
dic.academic.ru - academic Internet dictionary
mathematics.ru - educational site in mathematics
nSU.ru - Novosibirsk educational website state University
webmath.Ru - educational portal For students, applicants and schoolchildren.
exponenta.ru Educational Mathematical Site
ru.Tradimo.com - FREE online School trading
crypto.hut2.ru - Multidisciplinary information resource
poker-wiki.ru - Poker's free encyclopedia
sernam.ru - Scientific Library Selected natural scientific publications
reshim.su - Internet site by solving Tasks Control courses
uNFX.RU - Forex on UNFX: Training, Trading Signals, Trust
slovopedia.com - Big encyclopedic Dictionary Slopeadia
pokermansion.3dn.ru - Your guide in the world of poker
statanaliz.info - Information Blog " Statistical analysis Data "
forex trader.rf - Forex trader portal
megaFX.ru - Actual Analytics Forex
fX-BY.COM - all for trader
The mathematical expectation (middle value) of the random value of X, given on a discrete probabilistic space, is called the number M \u003d M [x] \u003d σx i p i, if the series converges absolutely.
Appointment of service. Using the service online calculated mathematical expectation, dispersion and rms deviation (See example). In addition, a graph of the distribution function F (X) is built.
The properties of the mathematical expectation of a random variable
- The mathematical expectation of a constant value is equal to her: M [C] \u003d C, C - constant;
- M \u003d C M [x]
- The mathematical expectation of the sum of random variables is equal to the sum of their mathematical expectations: m \u003d m [x] + m [y]
- The mathematical expectation of the product of independent random variables is equal to the product of their mathematical expectations: m \u003d m [x] m [y], if x and y are independent.
Properties of dispersion
- The dispersion of a constant value is zero: D (C) \u003d 0.
- A permanent multiplier can be discarded from under the sign of the dispersion, erecting it into the square: D (k * x) \u003d k 2 d (x).
- If the random variables x and y are independent, then the amount dispersion is equal to the amount of dispersions: D (x + y) \u003d d (x) + d (y).
- If the random variables x and y are dependent: D (x + y) \u003d dx + dy + 2 (x-m [x]) (y-m [y])
- Computational formula is valid for dispersion:
D (x) \u003d m (x 2) - (m (x)) 2
Example. Known mathematical expectations and dispersion of two independent random variables x and y: m (x) \u003d 8, m (y) \u003d 7, d (x) \u003d 9, d (y) \u003d 6. Find mathematical expectation and dispersion Random variance Z \u003d 9x-8Y + 7.
Decision. Based on the properties of the mathematical expectation: M (z) \u003d m (9x-8y + 7) \u003d 9 * m (x) - 8 * m (y) + m (7) \u003d 9 * 8 - 8 * 7 + 7 \u003d 23 .
Based on the properties of the dispersion: D (z) \u003d d (9x-8Y + 7) \u003d D (9x) - D (8Y) + D (7) \u003d 9 ^ 2D (x) - 8 ^ 2D (y) + 0 \u003d 81 * 9 - 64 * 6 \u003d 345
Algorithm for calculating mathematical expectation
Properties of discrete random variables: all their values \u200b\u200bcan be rented by natural numbers; Each value to compare the probability other than zero.- Alternately multiply the pairs: X i per P i.
- We fold the product of each pair x i p i.
For example, for n \u003d 4: m \u003d σx i p i \u003d x 1 p 1 + x 2 p 2 + x 3 p 3 + x 4 p 4
Example number 1.
| X I. | 1 | 3 | 4 | 7 | 9 |
| P I. | 0.1 | 0.2 | 0.1 | 0.3 | 0.3 |
The mathematical expectation is found according to the formula m \u003d σx i p i.
Mathematical expectation M [x].
M [x] \u003d 1 * 0.1 + 3 * 0.2 + 4 * 0.1 + 7 * 0.3 + 9 * 0.3 \u003d 5.9
The dispersion is found according to the formula d \u003d σx 2 i p i - m [x] 2.
Dispersion D [x].
D [x] \u003d 1 2 * 0.1 + 3 2 * 0.2 + 4 2 * 0.1 + 7 2 * 0.3 + 9 2 * 0.3 - 5.9 2 \u003d 7.69
Average quadratic deviation σ (x).
Σ \u003d SQRT (D [x]) \u003d SQRT (7.69) \u003d 2.78
Example number 2. The discrete random value has the following range of distribution:
| H. | -10 | -5 | 0 | 5 | 10 |
| r | but | 0,32 | 2a. | 0,41 | 0,03 |
Decision. The value of A find from the relation: Σp i \u003d 1
Σp i \u003d a + 0.32 + 2 A + 0.41 + 0.03 \u003d 0.76 + 3 a \u003d 1
0.76 + 3 a \u003d 1 or 0.24 \u003d 3 A, from where a \u003d 0.08
Example number 3. Determine the law of distribution of the discrete random variable, if its dispersion is known, and x 1
p 1 \u003d 0.3; P 2 \u003d 0.3; p 3 \u003d 0.1; P 4 \u003d 0.3
d (x) \u003d 12.96
Decision.
Here it is necessary to make a formula for finding dispersion D (x):
d (x) \u003d x 1 2 p 1 + x 2 2 p 2 + x 3 2 p 3 + x 4 2 p 4 -m (x) 2
where Miethazza M (x) \u003d x 1 p 1 + x 2 P 2 + x 3 p 3 + x 4 P 4
For our data
m (x) \u003d 6 * 0,3 + 9 * 0,3 + x 3 * 0.1 + 15 * 0.3 \u003d 9 + 0.1x 3
12.96 \u003d 6 2 0.3 + 9 2 0.3 + x 3 2 0.1 + 15 2 0.3- (9 + 0.1x 3) 2
or -9/100 (x 2 -20x + 96) \u003d 0
Accordingly, it is necessary to find the roots of the equation, and there will be two.
x 3 \u003d 8, x 3 \u003d 12
Choose the one that satisfies the condition X 1
Discrete random variable
x 1 \u003d 6; x 2 \u003d 9; x 3 \u003d 12; x 4 \u003d 15
p 1 \u003d 0.3; P 2 \u003d 0.3; p 3 \u003d 0.1; P 4 \u003d 0.3
There will be tasks for an independent solution to which you can see the answers.
Mathematical expectation and dispersion - most often applied numerical characteristics of a random variable. They characterize the most important distribution features: its position and the degree of dispersion. Mathematical expectation is often called simply medium value. random variable. Random variance - dispersion characteristic, random variance Near her mathematical expectation.
In many tasks, the practice is complete, an exhaustive characteristic of a random variable - the distribution law - or cannot be obtained, or not needed at all. In these cases, limited by an approximate description of a random variable using numerical characteristics.
Mathematical expectation of a discrete random variable
Let's come up to the concept of mathematical expectation. Let the mass of some substance distributed between the abscissa axis points x.1 , x.2 , ..., x.n.. In this case, each material point has a corresponding mass with a probability of p.1 , p.2 , ..., p.n.. It is required to choose one point on the abscissa axis, which characterizes the position of the entire system of material points, taking into account their masses. Naturally, as such a point, take the center of mass system of material points. This is the average weighted value of a random variable. X.which is the abscissa of each point x.i. It enters the "weight" equal to the corresponding probability. The average value of the random variable thus obtained X. It is called its mathematical expectation.
The mathematical expectation of the discrete random variable is the amount of works of all its possible values \u200b\u200bon the likelihood of these values:
Example 1. A win-win lottery is organized. There are 1000 winnings, of which 400 are 10 rubles. 300 - 20 rubles. 200 - 100 rubles. and 100 - 200 rubles. What is the average win size for bought one ticket?
Decision. The average winnings we will find if the total amount of winnings, which is equal to 10 * 400 + 20 * 300 + 100 * 200 + 200 * 100 \u003d 50,000 rubles, divide by 1000 (total winnings). Then we get 50000/1000 \u003d 50 rubles. But the expression for the average winning calculation can be represented as follows:
On the other hand, in these conditions, the amount of winnings is a random value that can take values \u200b\u200b10, 20, 100 and 200 rubles. with probabilities equal to 0.4, respectively; 0.3; 0.2; 0.1. Consequently, the expected average winnings is equal to the amount of product size of the winnings on the probability of their receipt.
Example 2. The publisher decided to publish a new book. It is going to sell the book for 280 rubles, of which 200 will receive himself, 50 - bookstore and 30 - author. The table provides information on the cost of publishing a book and the likelihood of selling a certain number of copies of the book.
Find the expected profit publisher.
Decision. Random magnitude "Profit" is equal to the difference in revenues from the sale and cost of costs. For example, if 500 copies of the book be sold, then the income from the sale is equal to 200 * 500 \u003d 100000, and the cost of edition is 225,000 rubles. Thus, the publisher threatens a loss of 125,000 rubles. The following table summarizes the expected values \u200b\u200bof the random variable - profits:
| Number | Profit x.i. | Probability p.i. | x.i. p.i. |
| 500 | -125000 | 0,20 | -25000 |
| 1000 | -50000 | 0,40 | -20000 |
| 2000 | 100000 | 0,25 | 25000 |
| 3000 | 250000 | 0,10 | 25000 |
| 4000 | 400000 | 0,05 | 20000 |
| Total: | 1,00 | 25000 |
Thus, we get a mathematical expectation of the publisher's profits:
.
Example 3. Probability of hitting one shot p. \u003d 0.2. Determine the flow charges that provide mathematical expectation of the number of hits equal to 5.
Decision. From the same formula for the expectation we used so far, express x. - Consumption of shells:
.
Example 4. Determine the mathematical expectation of a random variable x. the number of hits at three shots, if the probability of hitting each shot p. = 0,4 .
Tip: The likelihood of random values \u200b\u200bto find bernoulli formula .
Properties of mathematical expectation
Consider the properties of mathematical expectation.
Property 1.The mathematical expectation of a permanent value is equal to this constant:
Property 2.A permanent multiplier can be made for a sign of mathematical expectation:
Property 3.The mathematical expectation of the amount (difference) of random variables is equal to the amount (difference) of their mathematical expectations:
Property 4.The mathematical expectation of the work of random variables is equal to the product of their mathematical expectations:
Property 5.If all the values \u200b\u200bof the random variable X. Reduce (enlarge) on the same number FROMIt will reduce its mathematical expectation (will increase) on the same number:
When you can not be limited to mathematical expectation
In most cases, only a mathematical expectation cannot sufficiently characterize a random amount.
Let random variables X. and Y. specified by the following distribution laws:
| Value X. | Probability |
| -0,1 | 0,1 |
| -0,01 | 0,2 |
| 0 | 0,4 |
| 0,01 | 0,2 |
| 0,1 | 0,1 |
| Value Y. | Probability |
| -20 | 0,3 |
| -10 | 0,1 |
| 0 | 0,2 |
| 10 | 0,1 |
| 20 | 0,3 |
The mathematical expectations of these values \u200b\u200bare the same - zero is equal:
However, the nature of the distribution is different. Random value X. can take only values \u200b\u200bthat differ little from the mathematical expectation, but a random value Y. May take values \u200b\u200bsignificantly deviating from mathematical expectation. Similar example: average salary does not make it possible to judge the specific weight of highly and low-paid workers. In other words, according to mathematical expectation, it is impossible to judge what deviations from it, at least on average, are possible. To do this, you need to find a dispersion of a random variable.
Dispersion discrete random variable
Dispersion discrete random variable X. It is called the mathematical expectation of the square of its deviation from mathematical expectation:
Average quadratic deviation of random variable X. It is called the arithmetic value of the square root of its dispersion:
.
Example 5.Calculate dispersions and medium quadratic deviations of random variables X. and Y., whose distribution laws are shown in the tables above.
Decision. Mathematical expectations of random variables X. and Y.How was found above are zero. According to the dispersion formula E.(h.)=E.(y.) \u003d 0 Get:
Then the average quadratic deviations of random variables X. and Y. make up
.
Thus, with the same mathematical expectations of the dispersion of random variable X. very small, but a random variable Y. - Significant. This is a consequence of differences in their distribution.
Example 6. Investor has 4 alternative investment project. The table summarizes the data on the expected profit in these projects with an appropriate probability.
| Project 1. | Project 2. | Project 3. | Project 4. |
| 500, P.=1 | 1000, P.=0,5 | 500, P.=0,5 | 500, P.=0,5 |
| 0, P.=0,5 | 1000, P.=0,25 | 10500, P.=0,25 | |
| 0, P.=0,25 | 9500, P.=0,25 |
Find for each alternative mathematical expectation, dispersion and secondary quadratic deviation.
Decision. We show how these values \u200b\u200bare calculated for the 3rd alternative:
The table summarizes the found values \u200b\u200bfor all alternatives.
All alternatives are the same mathematical expectations. This means that in the long-term period, everyone has the same income. Standard deviation can be interpreted as a risk measurement unit - than it is more, the greater the risk of investment. An investor who does not want a big risk will choose a project 1, since he has the smallest standard deviation (0). If the investor prefers risk and greater revenues in a short period, he will choose the project with the greatest standard deviation - project 4.
Properties of dispersion
We give the properties of the dispersion.
Property 1.The dispersion of a constant value is zero:
Property 2.A permanent multiplier can be made for a dispersion sign, while placing it in the square:
.
Property 3.The dispersion of a random variable is equal to the mathematical expectation of the square of this value, from which the square of the mathematical expectation of the value is deducted:
,
where 
.
Property 4.The dispersion of the amount (difference) of random variables is equal to the amount (difference) of their dispersions:
Example 7. It is known that discrete random value X. It takes only two values: -3 and 7. In addition, a mathematical expectation is known: E.(X.) \u003d 4. Find dispersion of a discrete random variable.
Decision. Denote by p. the probability with which a random value takes the value x.1 = −3 . Then the probability of meaning x.2 = 7 will be 1 - p. . We derive an equation for mathematical expectation:
E.(X.) = x.1 p. + x.2 (1 − p.) = −3p. + 7(1 − p.) = 4 ,
where do you get probabilities: p. \u003d 0.3 and 1 - p. = 0,7 .
The law of the distribution of random variable:
| X. | −3 | 7 |
| p. | 0,3 | 0,7 |
The dispersion of this random variable is calculated by the formula from the dispersion properties 3:
D.(X.) = 2,7 + 34,3 − 16 = 21 .
Find a mathematical expectation of a random variable yourself, and then see the decision
Example 8. Discrete random variability X. Takes only two values. More from values \u200b\u200b3 it takes with a probability of 0.4. In addition, the dispersion of a random variable is known. D.(X.) \u003d 6. Find a mathematical expectation of a random variable.
Example 9. In the urn of 6 whites and 4 black balls. From the urns are taken out 3 balls. The number of white balls among the cuts of the balls is a discrete random variable X. . Find a mathematical expectation and dispersion of this random variable.
Decision. Random value X. can take values \u200b\u200b0, 1, 2, 3. The probability corresponding to them can be calculated by the rule of probability multiplication . The law of the distribution of random variable:
| X. | 0 | 1 | 2 | 3 |
| p. | 1/30 | 3/10 | 1/2 | 1/6 |
Hence the mathematical expectation of this random variable:
M.(X.) = 3/10 + 1 + 1/2 = 1,8 .
Dispersion of this random variable:
D.(X.) = 0,3 + 2 + 1,5 − 3,24 = 0,56 .
Mathematical expectation and dispersion of a continuous random variable
For a continuous random variable, the mechanical interpretation of the mathematical expectation will retain the same meaning: the center of mass for a single mass, distributed continuously on the abscissa axis with density f.(x.). Unlike a discrete random value, which has an argument function x.i. It changes hoppy, in a continuous random variable, the argument changes continuously. But the mathematical expectation of a continuous random variable is also associated with its average value.
To find a mathematical expectation and dispersion of a continuous random variable, you need to find certain integrals. . If the density function is given a continuous random variable, then it directly enters the integrand. If the probability distribution function is given, then, differentiating it, you need to find the density function.
The arithmetic average of all possible values \u200b\u200bof the continuous random variable is called it. mathematical expectationdenoted or.
In the theory of probability and in many of its applications, various numerical characteristics of random variables have great importance. The main ones are mathematical expectation and dispersion.
1. The mathematical expectation of the random variable and its properties.
Consider first the following example. Let the factory received a party consisting of N. Bearings. Wherein:
m 1. x 1,
m 2. - the number of bearings with an external diameter x 2,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
m N. - the number of bearings with an external diameter x N.,
Here m 1 + m 2 + ... + m n \u003d n. We find the arithmetic average x cf. external bearing diameter. Obviously
The outer diameter of the resulting impact of the bearing can be viewed as a random value that makes values x 1, x 2, ..., x N., C appropriate probabilities p 1 \u003d M 1 / N, p 2 \u003d M 2 / N, ..., p n \u003d m n / nsince the probability p I. The appearance of the bearing with an external diameter x I. equal m I / N. Thus, arithmetic meaning x cf. The outer diameter of the bearing can be determined by relation 
Let - a discrete random value with a given law of probability distribution
Values
x 1
x 2
. . .
x N.
Probability
p 1.
p 2.
. . .
p N.
Mathematical expectation discrete random variable The amount of paired works of all possible values \u200b\u200bof random variance to the probability corresponding to them, i.e. *
It is assumed that the immutable integral that stands in the right part of equality (40) exists.
Consider the properties of mathematical expectation. At the same time, we limit ourselves to the proof of only the first two properties, which will carry out for discrete random variables.
1 °. Mathematical expectation constant with equal to this constant.
Evidence. Permanent C. can be considered as a random value that can only take one value C. With a probability of equal unit. therefore 
2 °. Permanent multiplier can be made for a sign of mathematical expectation.. 
Evidence. Using the ratio (39), we have 
3 °. The mathematical expectation of the sum of several random variables is equal to the sum of the mathematical expectations of these values:
- The number of boys among 10 newborns.
It is quite clear that this amount is not known in advance, and in the next dozen children born, it may be:
Either boys - one and only one From the listed options.
And in order to keep the form, a little physical education:
- Long jump distance (in some units).
She is not able to predict even a master of sports :)
However, your hypotheses?
2) a continuous random value - takes everything Numeric values \u200b\u200bfrom some finite or infinite gap.
Note : In the educational literature, the abbreviations of DSV and NSV
First we will analyze the discrete random value, then - continuous.
Discrete random variable
- this is conformity between the possible values \u200b\u200bof this magnitude and their probabilities. Most often the law is recorded by the table: 
Quite often found term row
DistributionsBut in some situations he sounds ambiguous, and therefore I will adhere to the "law".
And now very important moment: since random value before Vick one of the meanings then the corresponding events form full group And the sum of probabilities of their occurrence is equal to one:
or, if you record it turns out:
For example, the law of the distribution of probabilities of the points falling on the cube is as follows: 
No comments.
Perhaps you have the impression that the discrete random value can take only "good" integer values. Let the illusion - they can be any:
Example 1.
Some game has the following Win distribution law: 
... Probably, you have long dreamed of such tasks :) I will reveal the secret - I, too. Especially after completed work on field Theory.
Decision: Since a random value can take only one of three values, then the corresponding events form full groupSo, the sum of their probabilities is equal to one: 
Explaining "Partizan": 
- Thus, the probability of winning conditional units is 0.4.
Control: what was required to make sure.
Answer:
It is not uncommon when the distribution law is required to be independently. For this use classical probability definition, theorems of multiplication / additions of events And other chips terver:
Example 2.
There are 50 lottery tickets in the box, among which are 12 winning, and 2 of them won 1000 rubles, and the rest are 100 rubles. Make the law of the distribution of a random variable - the winnings size, if one ticket is extracted from the box at random.
Decision: As you noticed, the values \u200b\u200bof the random variable are customary to be placed in order of their increase. Therefore, we start with the smallest winnings, and it is rubles.
Total plates 50 - 12 \u003d 38, and classical definition:
- The likelihood that the redemption received a learned ticket will be a bit.
With the rest of the case, everything is simple. The probability of winning rubles is:
Check: - And this is a particularly pleasant moment of such tasks!
Answer: The Second Win distribution law: 
The following task for self solutions:
Example 3.
The likelihood that the shooter will hit the target is equal to. Make the law of the distribution of a random variable - the number of hits after 2 shots.
... I knew that you missed him :) I remember multiplication and addition theorems. Solution and answer at the end of the lesson.
The distribution law fully describes a random amount, however, it is useful in practice (and sometimes more useful) to know only some of her numerical characteristics .
Mathematical expectation of a discrete random variable
In simple language, it is medium-priced value With multiple repetition of tests. Let a random value takes values \u200b\u200bwith probabilities 
respectively. Then the mathematical expectation of this random variable is equal amount of works All of its values \u200b\u200bon the corresponding probabilities:
or in the twisted form: 
Calculate, for example, the mathematical expectation of a random variable - the number of points falling on a playing cubicle:
Now let us remember our hypothetical game: 
The question arises: is it profitable to play this game at all? ... who have any impressions? So after all, the "Offhdka" and you can not say! But this question can be easily replied, computing the mathematical expectation, in fact - weighway In probabilities, winnings:
Thus, the mathematical expectation of this game losing.
Do not believe the impressions - truck!
Yes, here you can win 10 and even 20-30 times in a row, but on a long distance we are waiting for an inevitable ruin. And I would not advise you to play such games :) Well, maybe only for the sake of entertainment.
From the foregoing it follows that the mathematical expectation is no longer a random value.
Creative task for self-study:
Example 4.
Mr. X plays a European roulette on the following system: constantly puts 100 rubles to "Red". Make a law of the distribution of a random variable - his winnings. Calculate the mathematical wait of the winnings and roundate it to kopecks. how many average Loses a player with every hundreds supplied?
reference : European roulette contains 18 red, 18 black and 1 green sector (Zero). In the event of a "red" player, a twice rate is paid, otherwise it goes into casino income
There are many other roulette game systems for which you can make up your probability tables. But this is the case when we do not need any laws of distribution and table, because it is estimated that the mathematical expectation of the player will be exactly the same. From the system to the system is changing only
