The geometric meaning of mathematical expectation. Properties of mathematical expectation. Dispersion of a discrete random variable

Each individual value is completely determined by its distribution function. Also, to solve practical problems, it is enough to know several numerical characteristics, thanks to which it becomes possible to present the main features of a random variable in a concise form.

These quantities are primarily expected value and dispersion .

Expected value- the average value of a random variable in probability theory. Designated as .

by the most in a simple way mathematical expectation of a random variable X(w), are found as integralLebesgue with respect to the probability measure R initial probability space

You can also find the mathematical expectation of a value as Lebesgue integral from X by probability distribution R X quantities X:

where is the set of all possible values X.

Mathematical expectation of functions from a random variable X is through distribution R X. for instance, if X- random variable with values in and f(x)- unambiguous Borelfunction X , then:

If F(x)- distribution function X, then the mathematical expectation is representable integralLebesgue - Stieltjes (or Riemann - Stieltjes):

while the integrability X in what sense ( * ) corresponds to the finiteness of the integral

In specific cases, if X has a discrete distribution with probable values x k, k=1, 2, . , and probabilities , then

if X has an absolutely continuous distribution with a probability density p(x), then

in this case, the existence of a mathematical expectation is equivalent to the absolute convergence of the corresponding series or integral.

Properties of the mathematical expectation of a random variable.

The mathematical expectation of a constant value is equal to this value:

C- constant;

M=C.M[X]

The mathematical expectation of the sum of randomly taken values is equal to the sum of their mathematical expectations:

The mathematical expectation of the product of independent random variables = the product of their mathematical expectations:

M=M[X]+M[Y]

if X and Y independent.

if the series converges:

Algorithm for calculating the mathematical expectation.

Properties of discrete random variables: all their values can be renumbered natural numbers; equate each value with a non-zero probability.

1. Multiply the pairs in turn: x i on the pi.

2. Add the product of each pair x i p i.

For example, for n = 4 :

Distribution function of a discrete random variable stepwise, it increases abruptly at those points whose probabilities have a positive sign.

Example: Find the mathematical expectation by the formula.

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.

Service assignment. With an online service the 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 is a 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 dispersion of a constant value is equal to 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 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])
For the variance, the computational formula is valid:
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 dispersion properties: 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 the mathematical expectation

Properties of discrete random variables: all their values can be renumbered by natural numbers; Assign each value a non-zero probability.

Multiply the pairs one by one: x i by p i .
We 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

Distribution function of a discrete random variable stepwise, it increases abruptly at those points whose probabilities are positive.

Example #1.

x i	1	3	4	7	9
pi	0.1	0.2	0.1	0.3	0.3

The mathematical expectation is found by the formula m = ∑x i p i .
Mathematical expectation M[X].
M[x] = 1*0.1 + 3*0.2 + 4*0.1 + 7*0.3 + 9*0.3 = 5.9
The dispersion is found by the formula d = ∑x 2 i p i - M[x] 2 .
Dispersion 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 #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

Find the value a , the mathematical expectation and the standard deviation of this random variable.

Solution. The value 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 , whence a = 0.08

Example #3. Determine the distribution law of a discrete random variable if its variance is known, and x 1 x 1 =6; x2=9; x3=x; x4=15
p 1 =0.3; p2=0.3; p3=0.1; p 4 \u003d 0.3
d(x)=12.96

Solution.
Here you need to make 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 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, it is necessary to find the roots of the equation, and there will be two of them.
x 3 \u003d 8, x 3 \u003d 12
We choose the one that satisfies the condition x 1 x3=12

Distribution law of a discrete random variable
x 1 =6; x2=9; x 3 \u003d 12; x4=15
p 1 =0.3; p2=0.3; p3=0.1; p 4 \u003d 0.3

That is, if sl. quantity has a distribution law, then

called its mathematical expectation. If sl. the value has an infinite number of values, then the mathematical expectation is determined by the sum of an infinite series, provided that this series converges absolutely (otherwise, the mathematical expectation does not exist) .

For continuous sl. value given by the probability density function f(x), the mathematical expectation is determined as an integral

provided that this integral exists (if the integral diverges, then we say that the mathematical expectation does not exist).

Example 1. Let us define the mathematical expectation of a random variable distributed over Poisson's law. By definition

or denote

So the parameter , the defining distribution law of a Poisson random variable is equal to the mean value of this variable.

Example 2. For a random variable with an exponential distribution law, the mathematical expectation is

(in the integral, take the limits, taking into account the fact that f (x) is nonzero only for positive x).

Example 3. Random variable distributed according to the distribution law Cauchy, has no mean value. Really

Expectation Properties.

Property 1. The mathematical expectation of a constant is equal to the constant itself.

The constant C takes this value with a probability of one and, by definition, M(C)=C×1=C

Property 2. The mathematical expectation of the algebraic sum of random variables is equal to the algebraic sum of their mathematical expectations.

We confine ourselves to proving this property only for the sum of two discrete random variables, i.e. prove that

Under the sum of two discrete sl. The quantities are understood as Quantity that takes values with probabilities

By definition

where is the probability of the event calculated under the condition that . The right side of the last equality lists all cases of the occurrence of the event , therefore it is equal to the total probability of the occurrence of the event , i.e. . Likewise. Finally we have

Property 3. The mathematical expectation of the product of two independent random variables is equal to the product of their mathematical expectations.

At			…
Q			…
X			…
R			…

We give proofs of this property only for discrete quantities. For continuous random variables, it is proved similarly.

Let X and Y be independent and have distribution laws

The product of these random variables will be a random variable that takes values with probabilities equal, due to the independence of random variables, . Then

Consequence. The constant multiplier can be taken out of the mathematical expectation sign. So the century constant C does not depend on what value the next will take. value X, then by property 3. we have

M(CX)=M(C)×M(X)=C×M(X)

Example. If a and b are constants, then M(ax+b)=aM(x)+b.

Mathematical expectation of the number of occurrence of an event in the scheme of independent trials.

Let n independent experiments be performed, the probability of occurrence of an event in each of which is R. The number of occurrences of an event in these n experiments is a random variable X distributed according to the binomial law. However, the direct calculation of its average value is cumbersome. To simplify, we will use the decomposition, which we will use in the future repeatedly:

where has a distribution law (it takes the value 1 if the event occurred in the given experiment, and the value 0 if the event did not appear in the given experiment).


R	1st	R

those. the average number of occurrences of an event in n independent trials is equal to the product of the number of trials and the probability of occurrence of the event in one trial.

For example, if the probability of hitting the target with one shot is 0.1, then the average number of hits in 20 shots is 20×0.1=2.

The concept of mathematical expectation can be considered using the example of throwing a dice. With each throw, the dropped points are recorded. Natural values in the range 1 - 6 are used to express them.

After a certain number of throws, using simple calculations, you can find the arithmetic mean of the points that have fallen.

As well as dropping any of the range values, this value will be random.

And if you increase the number of throws several times? With a large number of throws, the arithmetic mean value of the points will approach a specific number, which in probability theory is called the mathematical expectation.

So, the mathematical expectation is understood as the average value of a random variable. This indicator can also be presented as a weighted sum of probable values.

This concept has several synonyms:

mean;
average value;
central trend indicator;
first moment.

In other words, it is nothing more than a number around which the values of a random variable are distributed.

In various spheres of human activity, approaches to understanding the mathematical expectation will be somewhat different.

It can be viewed as:

the average benefit received from the adoption of a decision, in the case when such a decision is considered from the point of view of the theory of large numbers;
the possible amount of winning or losing (gambling theory), calculated on average for each of the bets. In slang, they sound like "player's advantage" (positive for the player) or "casino advantage" (negative for the player);
percentage of profit received from winnings.

Mathematical expectation is not obligatory for absolutely all random variables. It is absent for those who have a discrepancy in the corresponding sum or integral.

Expectation Properties

Like any statistical parameter, mathematical expectation has the following properties:

Basic formulas for mathematical expectation

The calculation of the mathematical expectation can be performed both for random variables characterized by both continuity (formula A) and discreteness (formula B):

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 a given probability density.

Examples of calculating the mathematical expectation

Example A.

Is it possible to find out the average height of the gnomes in the fairy tale about Snow White. It is known that each of the 7 gnomes had a certain height: 1.25; 0.98; 1.05; 0.71; 0.56; 0.95 and 0.81 m.

The calculation algorithm is quite simple:

find the sum of all values of the growth indicator (random variable):
1,25+0,98+1,05+0,71+0,56+0,95+ 0,81 = 6,31;
The resulting amount is divided by the number of gnomes:
6,31:7=0,90.

Thus, the average height of gnomes in a fairy tale is 90 cm. In other words, this is the mathematical expectation of the growth of gnomes.

Working formula - M (x) \u003d 4 0.2 + 6 0.3 + 10 0.5 \u003d 6

Practical implementation of mathematical expectation

The calculation of a statistical indicator of mathematical expectation is resorted to in various fields of practical activity. First of all, we are talking about the commercial sphere. Indeed, the introduction of this indicator by Huygens is connected with the determination of the chances that can be favorable, or, on the contrary, unfavorable, for some event.

This parameter is widely used for risk assessment, especially when it comes to financial investments.
So, in business, the calculation of mathematical expectation acts as a method for assessing risk when calculating prices.

Also, this indicator can be used when calculating the effectiveness of certain measures, for example, on labor protection. Thanks to it, you can calculate the probability of an event occurring.

Another area of application of this parameter is management. It can also be calculated during product quality control. For example, using mat. expectations, you can calculate the possible number of manufacturing defective parts.

Mathematical expectation is also indispensable during the statistical processing of the results obtained in the course of scientific research. It also allows you to calculate the probability of a desired or undesirable outcome of an experiment or study, depending on the level of achievement of the goal. After all, its achievement can be associated with gain and profit, and its non-achievement - as a loss or loss.

Using Mathematical Expectation in Forex

The practical application of this statistical parameter is possible when conducting transactions in the foreign exchange market. It can be used to analyze the success of trade transactions. Moreover, an increase in the value of expectation indicates an increase in their success.

It is also important to remember that the mathematical expectation should not be considered as the only statistical parameter used to analyze the performance of a trader. The use of several statistical parameters along with the average value increases the accuracy of the analysis at times.

This parameter has proven itself well in monitoring observations of trading accounts. Thanks to him, a quick assessment of the work carried out on the deposit account is carried out. In cases where the trader's activity is successful and he avoids losses, it is not recommended to use only the calculation of mathematical expectation. In these cases, risks are not taken into account, which reduces the effectiveness of the analysis.

Conducted studies of traders' tactics indicate that:

the most effective are tactics based on random input;
the least effective are tactics based on structured inputs.

In order to achieve positive results, it is equally important:

money management tactics;
exit strategies.

Using such an indicator as the mathematical expectation, we can assume what will be the profit or loss when investing 1 dollar. It is known that this indicator, calculated for all games practiced in the casino, is in favor of the institution. This is what allows you to make money. In the case of a long series of games, the probability of losing money by the client increases significantly.

The games of professional players are limited to small time periods, which increases the chance of winning and reduces the risk of losing. The same pattern is observed in the performance of investment operations.

An investor can earn a significant amount with a positive expectation and a large number of transactions in a short time period.

Expectancy can be thought of as the difference between the percentage of profit (PW) times the average profit (AW) and the probability of loss (PL) times the average loss (AL).

As an example, consider the following: position - 12.5 thousand dollars, portfolio - 100 thousand dollars, risk per deposit - 1%. The profitability of transactions is 40% of cases with an average profit of 20%. In the event of a loss, the average loss is 5%. Calculating the mathematical expectation for a trade gives a value of $625.

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

The mathematical expectation, as will be shown below, is approximately equal to the average value of the random variable. To solve many problems, it is enough to know the mathematical expectation. For example, if it is known that the mathematical expectation of the number of points scored by the first shooter is greater than that of the second, then the first shooter, on average, knocks out more points than the second, and therefore shoots better than the second.

Definition 4.1: mathematical expectation A discrete random variable is called the sum of the products of all its possible values and their probabilities.

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

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

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

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

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

Solution: Random value X– number of occurrences of the event A has a Bernoulli distribution, so

In this way, the mathematical expectation of the number of occurrences of an event in one trial is equal to the probability of this event.

Probabilistic meaning of mathematical expectation

Let produced n tests in which the random variable X accepted m 1 times value x 1, m2 times value x2 ,…, m k times value x k, and m 1 + m 2 + …+ m k = n. Then the sum of all values taken X, is equal to x 1 m 1 + x 2 m 2 + …+ x k m k .

The arithmetic mean of all values taken by the random variable will be

Attitude m i / n- relative frequency Wi values x i approximately equal to the probability of occurrence of the event pi, where , That's why

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

Expectation Properties

Property1:The mathematical expectation of a constant value is equal to the constant itself

Property2:The constant factor can be taken out of the expectation sign

Definition 4.2: Two random variables called independent, if the distribution law of one of them does not depend on what possible values the other value has taken. Otherwise random variables are dependent.

Definition 4.3: Several random variables called mutually independent, if the distribution laws of any number of them do not depend on what possible values the other quantities have taken.

Property3:The mathematical expectation of the product of two independent random variables is equal to the product of their mathematical expectations.

Consequence:The mathematical expectation of the product of several mutually independent random variables is equal to the product of their mathematical expectations.

Property4:The mathematical expectation of the sum of two random variables is equal to the sum of their mathematical expectations.

Consequence:The mathematical expectation of the sum of several random variables is equal to the sum of their mathematical expectations.

Example. Calculate the mathematical expectation of a binomial random variable X- date of occurrence of the event A v n experiments.

Solution: Total number X event occurrences A in these trials is the sum of the number of occurrences of the event in the individual trials. We introduce random variables X i is the number of occurrences of the event in i th test, which are Bernoulli random variables with mathematical expectation , where . By the property of mathematical expectation, we have

In this way, the mean of the binomial distribution with parameters n and p is equal to the product of np.

Example. Probability of hitting a target when firing a gun p = 0.6. Find the mathematical expectation of the total number of hits if 10 shots are fired.

Solution: The hit at each shot does not depend on the outcomes of other shots, so the events under consideration are independent and, consequently, the desired mathematical expectation