– 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 numeric values from some finite or infinite interval.
Note : V educational literature popular abbreviations DSV and NSV
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 next law winning distribution: 
...you've probably dreamed of such tasks for a long time :) I'll tell you a secret - me too. Especially after finishing work on field theory.
Solution: since a random variable can only take one of three meanings, then 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. Create a distribution law 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 special nice moment such tasks!
Answer: the desired law of distribution of winnings: 
Next task for independent decision:
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
Speaking in simple language, This 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 and even 20-30 times in a row, but in the long run we will face inevitable ruin. 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 next system: 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 on average Does the player lose for every hundred he bet?
Reference : European roulette contains 18 red, 18 black and 1 green sector (“zero”). If “red” is rolled out, 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 and 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
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 most simple concepts theory of probability, 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 there's some going on random event, some kind of 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 classic definition this concept, you can begin to study mathematical expectation and variances of continuous random variables.
Arithmetic mean
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 is 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 the shift of values by equal value up or down. In addition, for independent tests the variance of the sum is equal to the sum of the variances.
Now we definitely need to consider examples of the dispersion 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.
Expectation
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 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. Such simple operations Not every quantity in probability theory allows you to do this. 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.
Another example
We ran 50 trials and got 10 types of outcomes - numbers from 0 to 9 - appearing in different percentage. 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 junior 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 in 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 normal distribution and want to see it directly square deviation, 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 horizontal axis and 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 educational institutions- it's 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.
In conclusion
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 scholarships.
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.
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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As is already known, the distribution law completely characterizes a random variable. However, often the distribution law is unknown and one has to limit oneself to less information. Sometimes it is even more profitable to use numbers that describe the random variable in total; such numbers are called numerical characteristics of a random variable. One of the important numerical characteristics is the mathematical expectation.
The mathematical expectation, as will be shown below, is approximately equal to the average value of the random variable. To solve many problems, it is enough to know the mathematical expectation. For example, if it is known that the mathematical expectation of the number of points scored by the first shooter is greater than that of the second, then the first shooter, on average, scores more points than the second, and, therefore, shoots better than the second. Although the mathematical expectation provides much less information about a random variable than the law of its distribution, knowledge of the mathematical expectation is sufficient for solving problems like the one above and many others.
§ 2. Mathematical expectation of a discrete random variable
Mathematical expectation A discrete random variable is the sum of the products of all its possible values and their probabilities.
Let the random variable X can only take values X 1 , X 2 , ..., X n , whose probabilities are respectively equal r 1 , r 2 , . . ., r n . Then the mathematical expectation M(X) random variable X is determined by equality
M(X) = X 1 r 1 + X 2 r 2 + … + x n p n .
If a discrete random variable X takes a countable set of possible values, then
M(X)=
Moreover, the mathematical expectation exists if the series on the right side of the equality converges absolutely.
Comment. From the definition it follows that the mathematical expectation of a discrete random variable is a non-random (constant) quantity. We recommend that you remember this statement, as it will be used many times later. It will be shown later that the mathematical expectation of a continuous random variable is also a constant value.
Example 1. Find the mathematical expectation of a random variable X, knowing the law of its distribution:
Solution. The required mathematical expectation is equal to the sum of the products of all possible values of the random variable and their probabilities:
M(X)= 3* 0, 1+ 5* 0, 6+ 2* 0, 3= 3, 9.
Example 2. Find the mathematical expectation of the number of occurrences of an event A in one trial, if the probability of the event A equal to r.
Solution. Random variable X - number of occurrences of the event A in one test - can take only two values: X 1 = 1 (event A occurred) with probability r And X 2 = 0 (event A did not occur) with probability q= 1 -r. The required mathematical expectation
M(X)= 1* p+ 0* q= p
So, the mathematical expectation of the number of occurrences of an event in one trial is equal to the probability of this event. This result will be used below.
§ 3. Probabilistic meaning of mathematical expectation
Let it be produced n tests in which the random variable X accepted T 1 times value X 1 , T 2 times value X 2 ,...,m k times value x k , and T 1 + T 2 + …+t To = p. Then the sum of all values taken X, equal to
X 1 T 1 + X 2 T 2 + ... + X To T To .
Let's find the arithmetic mean 
all values accepted by a random variable, for which we divide the found sum by the total number of tests:
=
(X 1 T 1
+
X 2 T 2 +
... +
X To T To)/p,
=
X 1
(m 1 /
n)
+
X 2
(m 2 /
n)
+ ... +
X To
(T To /n).
(*)
Noticing that the attitude m 1 / n- relative frequency W 1 values X 1 , m 2 / n - relative frequency W 2 values X 2 etc., we write the relation (*) like this:
=X 1
W 1
+
x 2 W 2
+ ..
. + X To W k .
(**)
Let us assume that the number of tests is large enough. Then the relative frequency is approximately equal to the probability of the event occurring (this will be proven in Chapter IX, § 6):
W 1
p 1 ,
W 2
p 2 ,
…,
W k
p k .
Replacing the relative frequencies with the corresponding probabilities in relation (**), we obtain
x 1 p 1
+
X 2 r 2
+ … +
X To r To .
The right side of this approximate equality is M(X). So,
M(X).
The probabilistic meaning of the result obtained is as follows: mathematical expectation is approximately equal(the more accurate, the greater the number of tests) the arithmetic mean of the observed values of a random variable.
Remark 1. It is easy to understand that the mathematical expectation is greater than the smallest and less than the largest possible value. In other words, on the number line, possible values are located to the left and right of the mathematical expectation. In this sense, the mathematical expectation characterizes the location of the distribution and is therefore often called distribution center.
This term is borrowed from mechanics: if the masses r 1 , p 2 , ..., r n located at the abscissa points x 1 ,
X 2 ,
...,
X n, and 
then the abscissa of the center of gravity
x c =
.
Considering that
=
M
(X)
And 
we get M(X)= x With .
So, the mathematical expectation is the abscissa of the center of gravity of a system of material points, the abscissas of which are equal to the possible values of the random variable, and the masses are equal to their probabilities.
Remark 2. The origin of the term “mathematical expectation” is associated with the initial period of the emergence of probability theory (XVI-XVII centuries), when the scope of its application was limited to gambling. The player was interested in the average value of the expected win, or, in other words, the mathematical expectation of winning.
01.02.2018
Mathematical expectation. 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.
Expectation- 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 (on 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 BO owners just have to wait, the market and negative checkmate expectations 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 approximately 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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