Basic numerical characteristics of discrete and continuous random variables: mathematical expectation, variance and standard deviation. Their properties and examples.
The distribution law (distribution function and distribution series or probability density) completely describes the behavior random variable. But in a number of problems, it is enough to know some numerical characteristics of the value under study (for example, its average value and possible deviation from it) in order to answer the question posed. Let's consider the main numerical characteristics of discrete random variables.
Definition 7.1.Mathematical expectation A discrete random variable is the sum of the products of its possible values and their corresponding probabilities:
M(X) = X 1 r 1 + X 2 r 2 + … + x p p p.(7.1)
If the number of possible values of a random variable is infinite, then if the resulting series converges absolutely.
Note 1. The mathematical expectation is sometimes called weighted average, since it is approximately equal to the arithmetic mean of the observed values of the random variable at large number experiments.
Note 2. From the definition of mathematical expectation it follows that its value is no less than the smallest possible value of a random variable and no more than the largest.
Note 3. The mathematical expectation of a discrete random variable is non-random(constant. We will see later that the same is true for continuous random variables.
Example 1. Find the mathematical expectation of a random variable X- the number of standard parts among three selected from a batch of 10 parts, including 2 defective ones. Let's create a distribution series for X. From the problem conditions it follows that X can take values 1, 2, 3. Then
Example 2. Determine the mathematical expectation of a random variable X- the number of coin tosses before the first appearance of the coat of arms. This quantity can take on an infinite number of values (the set of possible values is the set of natural numbers). Its distribution series has the form:
| X | … | n | … | ||
| r | 0,5 | (0,5) 2 | … | (0,5)n | … |
+ (when calculating, the formula for the sum of infinitely decreasing geometric progression: , where ).
Properties of mathematical expectation.
1) The mathematical expectation of a constant is equal to the constant itself:
M(WITH) = WITH.(7.2)
Proof. If we consider WITH as a discrete random variable taking only one value WITH with probability r= 1, then M(WITH) = WITH?1 = WITH.
2) The constant factor can be taken out of the sign of the mathematical expectation:
M(CX) = CM(X). (7.3)
Proof. If the random variable X given by distribution series
Then M(CX) = Cx 1 r 1 + Cx 2 r 2 + … + Cx p p p = WITH(X 1 r 1 + X 2 r 2 + … + x p r p) = CM(X).
Definition 7.2. Two random variables are called independent, if the distribution law of one of them does not depend on what values the other has taken. Otherwise the random variables dependent.
Definition 7.3. Let's call product of independent random variables X And Y random variable XY, the possible values of which are equal to the products of all possible values X for all possible values Y, and the corresponding probabilities are equal to the products of the probabilities of the factors.
3) The mathematical expectation of the product of two independent random variables is equal to the product of their mathematical expectations:
M(XY) = M(X)M(Y). (7.4)
Proof. To simplify calculations, we restrict ourselves to the case when X And Y take only two possible values:
Hence, M(XY) = x 1 y 1 ?p 1 g 1 + x 2 y 1 ?p 2 g 1 + x 1 y 2 ?p 1 g 2 + x 2 y 2 ?p 2 g 2 = y 1 g 1 (x 1 p 1 + x 2 p 2) + + y 2 g 2 (x 1 p 1 + x 2 p 2) = (y 1 g 1 + y 2 g 2) (x 1 p 1 + x 2 p 2) = M(X)?M(Y).
Note 1. This property can be proven similarly for a larger number of possible values of the factors.
Note 2. Property 3 is true for the product of any number of independent random variables, which is proven by mathematical induction.
Definition 7.4. Let's define sum of random variables X And Y as a random variable X+Y, the possible values of which are equal to the sums of each possible value X with every possible value Y; the probabilities of such sums are equal to the products of the probabilities of the terms (for dependent random variables - the products of the probability of one term by the conditional probability of the second).
4) The mathematical expectation of the sum of two random variables (dependent or independent) is equal to the sum of the mathematical expectations of the terms:
M (X+Y) = M (X) + M (Y). (7.5)
Proof.
Let us again consider the random variables defined by the distribution series given in the proof of property 3. Then the possible values X+Y are X 1 + at 1 , X 1 + at 2 , X 2 + at 1 , X 2 + at 2. Let us denote their probabilities respectively as r 11 , r 12 , r 21 and r 22. We'll find M(X+Y) = (x 1 + y 1)p 11 + (x 1 + y 2)p 12 + (x 2 + y 1)p 21 + (x 2 + y 2)p 22 =
= x 1 (p 11 + p 12) + x 2 (p 21 + p 22) + y 1 (p 11 + p 21) + y 2 (p 12 + p 22).
Let's prove that r 11 + r 22 = r 1. Indeed, the event that X+Y will take values X 1 + at 1 or X 1 + at 2 and the probability of which is r 11 + r 22 coincides with the event that X = X 1 (its probability is r 1). It is proved in a similar way that p 21 + p 22 = r 2 , p 11 + p 21 = g 1 , p 12 + p 22 = g 2. Means,
M(X+Y) = x 1 p 1 + x 2 p 2 + y 1 g 1 + y 2 g 2 = M (X) + M (Y).
Comment. From property 4 it follows that the sum of any number of random variables is equal to the sum of the mathematical expectations of the terms.
Example. Find the mathematical expectation of the sum of the number of points obtained when throwing five dice.
Let's find the mathematical expectation of the number of points rolled when throwing one dice:
M(X 1) = (1 + 2 + 3 + 4 + 5 + 6) The same number is equal to the mathematical expectation of the number of points rolled on any dice. Therefore, by property 4 M(X)=
Dispersion.
In order to have an idea of the behavior of a random variable, it is not enough to know only its mathematical expectation. Consider two random variables: X And Y, specified by distribution series of the form
| X | |||
| r | 0,1 | 0,8 | 0,1 |
| Y | ||
| p | 0,5 | 0,5 |
We'll find M(X) = 49?0,1 + 50?0,8 + 51?0,1 = 50, M(Y) = 0?0.5 + 100?0.5 = 50. As you can see, the mathematical expectations of both quantities are equal, but if for HM(X) well describes the behavior of a random variable, being its most probable possible value (and the remaining values do not differ much from 50), then the values Y significantly removed from M(Y). Therefore, along with the mathematical expectation, it is desirable to know how much the values of the random variable deviate from it. To characterize this indicator, dispersion is used.
Definition 7.5.Dispersion (scattering) of a random variable is the mathematical expectation of the square of its deviation from its mathematical expectation:
D(X) = M (X-M(X))². (7.6)
Let's find the variance of the random variable X(number of standard parts among those selected) in example 1 of this lecture. Let's calculate the squared deviation of each possible value from the mathematical expectation:
(1 - 2.4) 2 = 1.96; (2 - 2.4) 2 = 0.16; (3 - 2.4) 2 = 0.36. Hence,
Note 1. In determining dispersion, it is not the deviation from the mean itself that is assessed, but its square. This is done so that deviations of different signs do not cancel each other out.
Note 2. From the definition of dispersion it follows that this quantity takes only non-negative values.
Note 3. There is a formula for calculating variance that is more convenient for calculations, the validity of which is proven in the following theorem:
Theorem 7.1.D(X) = M(X²) - M²( X). (7.7)
Proof.
Using what M(X) is a constant value, and the properties of the mathematical expectation, we transform formula (7.6) to the form:
D(X) = M(X-M(X))² = M(X² - 2 X?M(X) + M²( X)) = M(X²) - 2 M(X)?M(X) + M²( X) =
= M(X²) - 2 M²( X) + M²( X) = M(X²) - M²( X), which was what needed to be proven.
Example. Let's calculate the variances of random variables X And Y discussed at the beginning of this section. M(X) = (49 2 ?0,1 + 50 2 ?0,8 + 51 2 ?0,1) - 50 2 = 2500,2 - 2500 = 0,2.
M(Y) = (0 2 ?0.5 + 100²?0.5) - 50² = 5000 - 2500 = 2500. So, the variance of the second random variable is several thousand times greater than the variance of the first. Thus, even without knowing the distribution laws of these quantities, based on the known dispersion values we can state that X deviates little from its mathematical expectation, while for Y this deviation is quite significant.
Properties of dispersion.
1) Variance of a constant value WITH equal to zero:
D (C) = 0. (7.8)
Proof. D(C) = M((C-M(C))²) = M((C-C)²) = M(0) = 0.
2) The constant factor can be taken out of the dispersion sign by squaring it:
D(CX) = C² D(X). (7.9)
Proof. D(CX) = M((CX-M(CX))²) = M((CX-CM(X))²) = M(C²( X-M(X))²) =
= C² D(X).
3) The variance of the sum of two independent random variables is equal to the sum of their variances:
D(X+Y) = D(X) + D(Y). (7.10)
Proof. D(X+Y) = M(X² + 2 XY + Y²) - ( M(X) + M(Y))² = M(X²) + 2 M(X)M(Y) +
+ M(Y²) - M²( X) - 2M(X)M(Y) - M²( Y) = (M(X²) - M²( X)) + (M(Y²) - M²( Y)) = D(X) + D(Y).
Corollary 1. The variance of the sum of several mutually independent random variables is equal to the sum of their variances.
Corollary 2. The variance of the sum of a constant and a random variable is equal to the variance of the random variable.
4) The variance of the difference between two independent random variables is equal to the sum of their variances:
D(X-Y) = D(X) + D(Y). (7.11)
Proof. D(X-Y) = D(X) + D(-Y) = D(X) + (-1)² D(Y) = D(X) + D(X).
The variance gives the average value of the squared deviation of a random variable from the mean; to evaluate the deviation itself, a value called the standard deviation is used.
Definition 7.6.Standard deviationσ random variable X called square root from dispersion:
Example. In the previous example, the standard deviations X And Y are equal respectively
The mathematical expectation (average value) of a random variable X given on a discrete probability space is the number m =M[X]=∑x i p i if the series converges absolutely.
Purpose of the service. Using the online service mathematical expectation, variance and standard deviation are calculated(see example). In addition, a graph of the distribution function F(X) is plotted.
Properties of the mathematical expectation of a random variable
- The mathematical expectation of a constant value is equal to itself: M[C]=C, C – constant;
- M=C M[X]
- The mathematical expectation of the sum of random variables is equal to the sum of their mathematical expectations: M=M[X]+M[Y]
- The mathematical expectation of the product of independent random variables is equal to the product of their mathematical expectations: M=M[X] M[Y] , if X and Y are independent.
Dispersion properties
- The variance of a constant value is zero: D(c)=0.
- The constant factor can be taken out from under the dispersion sign by squaring it: D(k*X)= k 2 D(X).
- If the random variables X and Y are independent, then the variance of the sum is equal to the sum of the variances: D(X+Y)=D(X)+D(Y).
- If random variables X and Y are dependent: D(X+Y)=DX+DY+2(X-M[X])(Y-M[Y])
- The following computational formula is valid for dispersion:
D(X)=M(X 2)-(M(X)) 2
Example. The mathematical expectations and variances of two independent random variables X and Y are known: M(x)=8, M(Y)=7, D(X)=9, D(Y)=6. Find the mathematical expectation and variance of the random variable Z=9X-8Y+7.
Solution. Based on the properties of mathematical expectation: M(Z) = M(9X-8Y+7) = 9*M(X) - 8*M(Y) + M(7) = 9*8 - 8*7 + 7 = 23 .
Based on the properties of dispersion: D(Z) = D(9X-8Y+7) = D(9X) - D(8Y) + D(7) = 9^2D(X) - 8^2D(Y) + 0 = 81*9 - 64*6 = 345
Algorithm for calculating mathematical expectation
Properties of discrete random variables: all their values can be renumbered natural numbers; each value is associated with a non-zero probability.- We multiply the pairs one by one: x i by p i .
- Add the product of each pair x i p i .
For example, for n = 4: m = ∑x i p i = x 1 p 1 + x 2 p 2 + x 3 p 3 + x 4 p 4
Example No. 1.
| x i | 1 | 3 | 4 | 7 | 9 |
| p i | 0.1 | 0.2 | 0.1 | 0.3 | 0.3 |
We find the mathematical expectation using the formula m = ∑x i p i .
Expectation M[X].
M[x] = 1*0.1 + 3*0.2 + 4*0.1 + 7*0.3 + 9*0.3 = 5.9
We find the variance using the formula d = ∑x 2 i p i - M[x] 2 .
Variance D[X].
D[X] = 1 2 *0.1 + 3 2 *0.2 + 4 2 *0.1 + 7 2 *0.3 + 9 2 *0.3 - 5.9 2 = 7.69
Standard deviation σ(x).
σ = sqrt(D[X]) = sqrt(7.69) = 2.78
Example No. 2. A discrete random variable has the following distribution series:
| X | -10 | -5 | 0 | 5 | 10 |
| r | A | 0,32 | 2a | 0,41 | 0,03 |
Solution. The value of a is found from the relation: Σp i = 1
Σp i = a + 0.32 + 2 a + 0.41 + 0.03 = 0.76 + 3 a = 1
0.76 + 3 a = 1 or 0.24=3 a , from where a = 0.08
Example No. 3. Determine the distribution law of a discrete random variable if its variance is known, and x 1
p 1 =0.3; p 2 =0.3; p 3 =0.1; p 4 =0.3
d(x)=12.96
Solution.
Here you need to create a formula for finding the variance d(x):
d(x) = x 1 2 p 1 +x 2 2 p 2 +x 3 2 p 3 +x 4 2 p 4 -m(x) 2
where the expectation m(x)=x 1 p 1 +x 2 p 2 +x 3 p 3 +x 4 p 4
For our data
m(x)=6*0.3+9*0.3+x 3 *0.1+15*0.3=9+0.1x 3
12.96 = 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)=0
Accordingly, we need to find the roots of the equation, and there will be two of them.
x 3 =8, x 3 =12
Choose the one that satisfies the condition x 1
Distribution law of a discrete random variable
x 1 =6; x 2 =9; x 3 =12; x 4 =15
p 1 =0.3; p 2 =0.3; p 3 =0.1; p 4 =0.3
In probability theory and in many of its applications, various numerical characteristics of random variables are of great importance. The main ones are mathematical expectation and variance.
1. Mathematical expectation of a random variable and its properties.
Let's first consider the following example. Let the plant receive a batch consisting of N bearings. In this case:
m 1 x 1,
m 2- number of bearings with outer diameter x 2,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
m n- number of bearings with outer diameter x n,
Here m 1 +m 2 +...+m n =N. Let's find the arithmetic mean x avg outer diameter of the bearing. Obviously,
The outer diameter of a bearing taken out at random can be considered as a random variable taking the values x 1, x 2, ..., x n, with corresponding probabilities p 1 =m 1 /N, p 2 =m 2 /N, ..., p n =m n /N, since the probability p i appearance of a bearing with an outer diameter x i equal to m i /N. Thus, the arithmetic mean x avg The outer diameter of the bearing can be determined using the relation 
Let be a discrete random variable with a given probability distribution law
Values
x 1
x 2
. . .
x n
Probabilities
p 1
p2
. . .
p n
Mathematical expectation discrete random variable is the sum of paired products of all possible values of a random variable by their corresponding probabilities, i.e. *
In this case, it is assumed that the improper integral on the right side of equality (40) exists.
Let's consider the properties of mathematical expectation. In this case, we will limit ourselves to the proof of only the first two properties, which we will carry out for discrete random variables.
1°. The mathematical expectation of the constant C is equal to this constant.
Proof. Constant C can be thought of as a random variable that can only take one value C with probability equal to one. That's why 
2°. The constant factor can be taken beyond the sign of the mathematical expectation, i.e. 
Proof. Using relation (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 variables:
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
The concept of mathematical expectation can be considered using the example of throwing a die. With each throw, the dropped points are recorded. To express them, natural values in the range 1 – 6 are used.
After a certain number of throws, using simple calculations, you can find the arithmetic average of the points rolled.
Just like the occurrence of any of the values in the range, this value will be random.
What if you increase the number of throws several times? With a large number of throws, the arithmetic average of the points will approach a specific number, which in probability theory is called the mathematical expectation.
So, by mathematical expectation we mean the average value of a random variable. This indicator can also be presented as a weighted sum of probable value values.
This concept has several synonyms:
- average value;
- average value;
- indicator of central tendency;
- first moment.
In other words, it is nothing more than a number around which the values of a random variable are distributed.
In different spheres of human activity, approaches to understanding mathematical expectation will be somewhat different.
It can be considered as:
- the average benefit obtained from making a decision, when such a decision is considered from the point of view of large number theory;
- the possible amount of winning or losing (gambling theory), calculated on average for each bet. In slang, they sound like “player’s advantage” (positive for the player) or “casino advantage” (negative for the player);
- percentage of profit received from winnings.
The expectation is not mandatory for absolutely all random variables. It is absent for those who have a discrepancy in the corresponding sum or integral.
Properties of mathematical expectation
Like any statistical parameter, the mathematical expectation has the following properties:
Basic formulas for mathematical expectation
The calculation of the mathematical expectation can be performed both for random variables characterized by both continuity (formula A) and discreteness (formula B):
- M(X)=∑i=1nxi⋅pi, where xi are the values of the random variable, pi are the probabilities:
- M(X)=∫+∞−∞f(x)⋅xdx, where f(x) is the given probability density.
Examples of calculating mathematical expectation
Example A.
Is it possible to find out the average height of the dwarves in the fairy tale about Snow White. It is known that each of the 7 dwarves had a certain height: 1.25; 0.98; 1.05; 0.71; 0.56; 0.95 and 0.81 m.
The calculation algorithm is quite simple:
- we find the sum of all values of the growth indicator (random variable):
1,25+0,98+1,05+0,71+0,56+0,95+ 0,81 = 6,31; - Divide the resulting amount by the number of gnomes:
6,31:7=0,90.
Thus, the average height of gnomes in a fairy tale is 90 cm. In other words, this is the mathematical expectation of the growth of gnomes.
Working formula - M(x)=4 0.2+6 0.3+10 0.5=6
Practical implementation of mathematical expectation
The calculation of the statistical indicator of mathematical expectation is resorted to in various areas of practical activity. First of all, we are talking about the commercial sphere. After all, Huygens’s introduction of this indicator is associated with determining the chances that can be favorable, or, on the contrary, unfavorable, for some event.
This parameter is widely used to assess risks, especially when it comes to financial investments.
Thus, in business, the calculation of mathematical expectation acts as a method for assessing risk when calculating prices.
This indicator can also be used to calculate the effectiveness of certain measures, for example, labor protection. Thanks to it, you can calculate the probability of an event occurring.
Another area of application of this parameter is management. It can also be calculated during product quality control. For example, using mat. expectations, you can calculate the possible number of defective parts produced.
The mathematical expectation also turns out to be indispensable when carrying out statistical processing of the results obtained during scientific research. It allows you to calculate the probability of a desired or undesirable outcome of an experiment or study depending on the level of achievement of the goal. After all, its achievement can be associated with gain and benefit, and its failure can be associated with loss or loss.
Using mathematical expectation in Forex
The practical application of this statistical parameter is possible when conducting operations on the foreign exchange market. With its help, you can analyze the success of trade transactions. Moreover, an increase in the expectation value indicates an increase in their success.
It is also important to remember that the mathematical expectation should not be considered as the only statistical parameter used to analyze a trader’s performance. The use of several statistical parameters along with the average value increases the accuracy of the analysis significantly.
This parameter has proven itself well in monitoring observations of trading accounts. Thanks to it, a quick assessment of the work carried out on the deposit account is carried out. In cases where the trader’s activity is successful and he avoids losses, it is not recommended to use exclusively the calculation of mathematical expectation. In these cases, risks are not taken into account, which reduces the effectiveness of the analysis.
Conducted studies of traders’ tactics indicate that:
- The most effective tactics are those based on random entry;
- The least effective are tactics based on structured inputs.
In achieving positive results, no less important are:
- money management tactics;
- exit strategies.
Using such an indicator as the mathematical expectation, you can predict what the profit or loss will be when investing 1 dollar. It is known that this indicator, calculated for all games practiced in the casino, is in favor of the establishment. This is what allows you to make money. In the case of a long series of games, the likelihood of a client losing money increases significantly.
Games played by professional players are limited to short periods of time, which increases the likelihood of winning and reduces the risk of losing. The same pattern is observed when performing investment operations.
An investor can earn a significant amount by having positive expectations and making a large number of transactions in a short period of time.
Expectation can be thought of as the difference between the percentage of profit (PW) multiplied by the average profit (AW) and the probability of loss (PL) multiplied by the average loss (AL).
As an example, consider the following: position – 12.5 thousand dollars, portfolio – 100 thousand dollars, deposit risk – 1%. The profitability of transactions is 40% of cases with an average profit of 20%. In case of loss, the average loss is 5%. Calculating the mathematical expectation for the transaction gives a value of $625.
