The formula for calculating the mathematical expectation of a discrete random variable. Properties of mathematical expectation. Mathematical expectation when playing poker

Chapter 6

Numerical characteristics of random variables

Mathematical expectation and its properties

To solve many practical problems, it is not always necessary to know all possible values ​​of a random variable and their probabilities. Moreover, sometimes the distribution law of the random variable under study is simply unknown. However, it is required to highlight some features of this random variable, in other words, numerical characteristics.

Numerical characteristics- these are some numbers characterizing certain properties, distinctive features of a random variable.

For example, the average value of a random variable, the average spread of all values ​​of a random variable around its average, etc. The main purpose of numerical characteristics is to express in a concise form the most important features of the distribution of the random variable under study. Numerical characteristics in probability theory play a huge role. They help to solve, even without knowledge of distribution laws, many important practical problems.

Among all the numerical characteristics, first of all, we single out position characteristics. These are characteristics that fix the position of a random variable on the number axis, i.e. a certain average value, around which the remaining values ​​of the random variable are grouped.

Of the characteristics of the position, the mathematical expectation plays the greatest role in probability theory.

Expected value sometimes referred to simply as the mean value of a random variable. It is a kind of distribution center.

Mathematical expectation of a discrete random variable

Consider the concept of mathematical expectation first for a discrete random variable.

Before introducing a formal definition, we solve the following simple problem.

Example 6.1. Let a shooter fire 100 shots at a target. As a result, the following picture was obtained: 50 shots - hitting the "eight", 20 shots - hitting the "nine" and 30 - hitting the "ten". What is the average score per shot.

Decision of this problem is obvious and comes down to finding the average value of 100 numbers, namely points.

We transform the fraction by dividing the numerator by the denominator term by term, and represent the average value in the form of the following formula:

Let us now assume that the number of points in one shot is the values ​​of some discrete random variable X. It is clear from the condition of the problem that X 1 =8; X 2 =9; X 3=10. The relative frequencies of occurrence of these values ​​are known, which, as is known, are approximately equal to the probabilities of the corresponding values ​​for a large number of tests, i.e. R 1 ≈0,5;R 2 ≈0,2; R 3 ≈0.3. So, . The value on the right side is the mathematical expectation of a discrete random variable.

Mathematical expectation of a discrete random variable X is the sum of the products of all its possible values ​​and the probabilities of these values.

Let a discrete random variable X given by its distribution series:

Then the mathematical expectation M(X) of a discrete random variable is determined by the following formula:

If a discrete random variable takes on an infinite countable set of values, then the mathematical expectation is expressed by the formula:

,

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

Example 6.2 . Find the mathematical expectation of winning X under the conditions of example 5.1.

Decision . Recall that the distribution series X has the following form:

Get M(X)=0∙0.7+10∙0.2+50∙0.1=7. Obviously, 7 rubles is the fair price of a ticket in this lottery, without various costs, for example, associated with the distribution or production of tickets. ■

Example 6.3 . Let the random variable X is the number of occurrences of some event BUT in one test. The probability of this event is R. To find M(X).

Decision. Obviously, the possible values ​​of the random variable are: X 1 =0 - event BUT did not appear and X 2 =1 – event BUT appeared. The distribution series has the form:

Then M(X) = 0∙(1−R)+1∙R= R. ■

So, the mathematical expectation of the number of occurrences of an event in one test is equal to the probability of this event.

At the beginning of the paragraph, a specific problem was given, where the relationship between the mathematical expectation and the average value of a random variable was indicated. Let us explain this in a general way.

Let produced k tests in which the random variable X accepted k 1 time value X 1 ; k 2 times value X 2 etc. and finally k n times value x n . It's obvious that k 1 +k 2 +…+k n = k. Let's find the arithmetic mean of all these values, we have

Note that the fraction is the relative frequency of occurrence of the value x i in k tests. With a large number of tests, the relative frequency is approximately equal to the probability, i.e. . Hence it follows that

.

Thus, the mathematical expectation is approximately equal to the arithmetic mean of the observed values ​​of a random variable, and the more accurate the greater the number of trials - this is probabilistic meaning of mathematical expectation.

The mathematical expectation is sometimes called center distribution of a random variable, since it is obvious that the possible values ​​of a random variable are located on the numerical axis to the left and to the right of its mathematical expectation.

Let us now turn to the concept of mathematical expectation for a continuous random variable.

Random variable a variable is called which, as a result of each test, takes on one previously unknown value, depending on random causes. Random variables are denoted by capital Latin letters: $X,\ Y,\ Z,\ \dots $ By their type, random variables can be discrete and continuous.

Discrete random variable- this is such a random variable, the values ​​​​of which can be no more than countable, that is, either finite or countable. Countability means that the values ​​of a random variable can be enumerated.

Example 1 . Let us give examples of discrete random variables:

a) the number of hits on the target with $n$ shots, here the possible values ​​are $0,\ 1,\ \dots ,\ n$.

b) the number of coats of arms that fell out when tossing a coin, here the possible values ​​are $0,\ 1,\ \dots ,\ n$.

c) the number of ships that arrived on board (a countable set of values).

d) the number of calls arriving at the exchange (a countable set of values).

1. Law of probability distribution of a discrete random variable.

A discrete random variable $X$ can take the values ​​$x_1,\dots ,\ x_n$ with probabilities $p\left(x_1\right),\ \dots ,\ p\left(x_n\right)$. The correspondence between these values ​​and their probabilities is called distribution law of a discrete random variable. As a rule, this correspondence is specified using a table, in the first line of which the values ​​of $x_1,\dots ,\ x_n$ are indicated, and in the second line the probabilities corresponding to these values ​​are $p_1,\dots ,\ p_n$.

$\begin(array)(|c|c|)
\hline
X_i & x_1 & x_2 & \dots & x_n \\
\hline
p_i & p_1 & p_2 & \dots & p_n \\
\hline
\end(array)$

Example 2 . Let the random variable $X$ be the number of points rolled when a dice is rolled. Such a random variable $X$ can take the following values ​​$1,\ 2,\ 3,\ 4,\ 5,\ 6$. The probabilities of all these values ​​are equal to $1/6$. Then the probability distribution law for the random variable $X$:

$\begin(array)(|c|c|)
\hline
1 & 2 & 3 & 4 & 5 & 6 \\
\hline

\hline
\end(array)$

Comment. Since the events $1,\ 2,\ \dots ,\ 6$ form a complete group of events in the distribution law of the discrete random variable $X$, the sum of the probabilities must be equal to one, i.e. $\sum(p_i)=1$.

2. Mathematical expectation of a discrete random variable.

Mathematical expectation of a random variable specifies its "central" value. For a discrete random variable, the mathematical expectation is calculated as the sum of the products of the values ​​$x_1,\dots ,\ x_n$ and the probabilities $p_1,\dots ,\ p_n$ corresponding to these values, i.e.: $M\left(X\right)=\sum ^n_(i=1)(p_ix_i)$. In English literature, another notation $E\left(X\right)$ is used.

Expectation Properties$M\left(X\right)$:

1. $M\left(X\right)$ is between the smallest and largest values ​​of the random variable $X$.
2. The mathematical expectation of a constant is equal to the constant itself, i.e. $M\left(C\right)=C$.
3. The constant factor can be taken out of the expectation sign: $M\left(CX\right)=CM\left(X\right)$.
4. The mathematical expectation of the sum of random variables is equal to the sum of their mathematical expectations: $M\left(X+Y\right)=M\left(X\right)+M\left(Y\right)$.
5. The mathematical expectation of the product of independent random variables is equal to the product of their mathematical expectations: $M\left(XY\right)=M\left(X\right)M\left(Y\right)$.

Example 3 . Let's find the mathematical expectation of the random variable $X$ from example $2$.

$$M\left(X\right)=\sum^n_(i=1)(p_ix_i)=1\cdot ((1)\over (6))+2\cdot ((1)\over (6) )+3\cdot ((1)\over (6))+4\cdot ((1)\over (6))+5\cdot ((1)\over (6))+6\cdot ((1 )\over (6))=3.5.$$

We can notice that $M\left(X\right)$ is between the smallest ($1$) and largest ($6$) values ​​of the random variable $X$.

Example 4 . It is known that the mathematical expectation of the random variable $X$ is equal to $M\left(X\right)=2$. Find the mathematical expectation of the random variable $3X+5$.

Using the above properties, we get $M\left(3X+5\right)=M\left(3X\right)+M\left(5\right)=3M\left(X\right)+5=3\cdot 2 +5=11$.

Example 5 . It is known that the mathematical expectation of the random variable $X$ is equal to $M\left(X\right)=4$. Find the mathematical expectation of the random variable $2X-9$.

Using the above properties, we get $M\left(2X-9\right)=M\left(2X\right)-M\left(9\right)=2M\left(X\right)-9=2\cdot 4 -9=-1$.

3. Dispersion of a discrete random variable.

Possible values ​​of random variables with equal mathematical expectations can scatter differently around their average values. For example, in two student groups, the average score for the exam in probability theory turned out to be 4, but in one group everyone turned out to be good students, and in the other group - only C students and excellent students. Therefore, there is a need for such a numerical characteristic of a random variable, which would show the spread of the values ​​of a random variable around its mathematical expectation. This characteristic is dispersion.

Dispersion of a discrete random variable$X$ is:

$$D\left(X\right)=\sum^n_(i=1)(p_i(\left(x_i-M\left(X\right)\right))^2).\ $$

In English literature, the notation $V\left(X\right),\ Var\left(X\right)$ is used. Very often the variance $D\left(X\right)$ is calculated by the formula $D\left(X\right)=\sum^n_(i=1)(p_ix^2_i)-(\left(M\left(X \right)\right))^2$.

Dispersion Properties$D\left(X\right)$:

1. The dispersion is always greater than or equal to zero, i.e. $D\left(X\right)\ge 0$.
2. The dispersion from a constant is equal to zero, i.e. $D\left(C\right)=0$.
3. The constant factor can be taken out of the dispersion sign, provided that it is squared, i.e. $D\left(CX\right)=C^2D\left(X\right)$.
4. The variance of the sum of independent random variables is equal to the sum of their variances, i.e. $D\left(X+Y\right)=D\left(X\right)+D\left(Y\right)$.
5. The variance of the difference of independent random variables is equal to the sum of their variances, i.e. $D\left(X-Y\right)=D\left(X\right)+D\left(Y\right)$.

Example 6 . Let us calculate the variance of the random variable $X$ from example $2$.

$$D\left(X\right)=\sum^n_(i=1)(p_i(\left(x_i-M\left(X\right)\right))^2)=((1)\over (6))\cdot (\left(1-3,5\right))^2+((1)\over (6))\cdot (\left(2-3,5\right))^2+ \dots +((1)\over (6))\cdot (\left(6-3,5\right))^2=((35)\over (12))\approx 2.92.$$

Example 7 . It is known that the variance of the random variable $X$ is equal to $D\left(X\right)=2$. Find the variance of the random variable $4X+1$.

Using the above properties, we find $D\left(4X+1\right)=D\left(4X\right)+D\left(1\right)=4^2D\left(X\right)+0=16D\ left(X\right)=16\cdot 2=32$.

Example 8 . It is known that the variance of $X$ is equal to $D\left(X\right)=3$. Find the variance of the random variable $3-2X$.

Using the above properties, we find $D\left(3-2X\right)=D\left(3\right)+D\left(2X\right)=0+2^2D\left(X\right)=4D\ left(X\right)=4\cdot 3=12$.

4. Distribution function of a discrete random variable.

The method of representing a discrete random variable in the form of a distribution series is not the only one, and most importantly, it is not universal, since a continuous random variable cannot be specified using a distribution series. There is another way to represent a random variable - the distribution function.

distribution function random variable $X$ is a function $F\left(x\right)$, which determines the probability that the random variable $X$ takes a value less than some fixed value $x$, i.e. $F\left(x\right)$ )=P\left(X< x\right)$

Distribution function properties:

1. $0\le F\left(x\right)\le 1$.
2. The probability that the random variable $X$ takes values ​​from the interval $\left(\alpha ;\ \beta \right)$ is equal to the difference between the values ​​of the distribution function at the ends of this interval: $P\left(\alpha< X < \beta \right)=F\left(\beta \right)-F\left(\alpha \right)$
3. $F\left(x\right)$ - non-decreasing.
4. $(\mathop(lim)_(x\to -\infty ) F\left(x\right)=0\ ),\ (\mathop(lim)_(x\to +\infty ) F\left(x \right)=1\ )$.

Example 9 . Let us find the distribution function $F\left(x\right)$ for the distribution law of the discrete random variable $X$ from example $2$.

$\begin(array)(|c|c|)
\hline
1 & 2 & 3 & 4 & 5 & 6 \\
\hline
1/6 & 1/6 & 1/6 & 1/6 & 1/6 & 1/6 \\
\hline
\end(array)$

If $x\le 1$, then obviously $F\left(x\right)=0$ (including $x=1$ $F\left(1\right)=P\left(X< 1\right)=0$).

If $1< x\le 2$, то $F\left(x\right)=P\left(X=1\right)=1/6$.

If $2< x\le 3$, то $F\left(x\right)=P\left(X=1\right)+P\left(X=2\right)=1/6+1/6=1/3$.

If $3< x\le 4$, то $F\left(x\right)=P\left(X=1\right)+P\left(X=2\right)+P\left(X=3\right)=1/6+1/6+1/6=1/2$.

If $4< x\le 5$, то $F\left(X\right)=P\left(X=1\right)+P\left(X=2\right)+P\left(X=3\right)+P\left(X=4\right)=1/6+1/6+1/6+1/6=2/3$.

If $5< x\le 6$, то $F\left(x\right)=P\left(X=1\right)+P\left(X=2\right)+P\left(X=3\right)+P\left(X=4\right)+P\left(X=5\right)=1/6+1/6+1/6+1/6+1/6=5/6$.

If $x > 6$ then $F\left(x\right)=P\left(X=1\right)+P\left(X=2\right)+P\left(X=3\right) +P\left(X=4\right)+P\left(X=5\right)+P\left(X=6\right)=1/6+1/6+1/6+1/6+ 1/6+1/6=1$.

So $F(x)=\left\(\begin(matrix)
0,\ at\ x\le 1,\\
1/6, at \ 1< x\le 2,\\
1/3,\ at\ 2< x\le 3,\\
1/2, at \ 3< x\le 4,\\
2/3,\ at\ 4< x\le 5,\\
5/6, \ at \ 4< x\le 5,\\
1,\ for \ x > 6.
\end(matrix)\right.$

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 has received the name of 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):

1. M(X)=∑i=1nxi⋅pi, where xi are the values ​​of the random variable, pi are the probabilities:
2. 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.

Decision:

6.1.2 Expectation properties

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

2. A constant factor can be taken out of the expectation sign.

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

This property is valid for an arbitrary number of random variables.

4. The mathematical expectation of the sum of two random variables is equal to the sum of the mathematical expectations of the terms.

This property is also true for an arbitrary number of random variables.

Example: M(X) = 5, M(Y)= 2. Find the mathematical expectation of a random variable Z, applying the properties of mathematical expectation, if it is known that Z=2X + 3Y.

Decision: M(Z) = M(2X + 3Y) = M(2X) + M(3Y) = 2M(X) + 3M(Y) = 2∙5+3∙2 =

1) the mathematical expectation of the sum is equal to the sum of the mathematical expectations

2) the constant factor can be taken out of the expectation sign

Let n independent trials be performed, the probability of occurrence of event A in which is equal to p. Then the following theorem holds:

Theorem. The mathematical expectation M(X) of the number of occurrences of event A in n independent trials is equal to the product of the number of trials and the probability of occurrence of the event in each trial.

6.1.3 Dispersion of a discrete random variable

Mathematical expectation cannot fully characterize a random process. In addition to the mathematical expectation, it is necessary to introduce a value that characterizes the deviation of the values ​​of the random variable from the mathematical expectation.

This deviation is equal to the difference between the random variable and its mathematical expectation. In this case, the mathematical expectation of the deviation is zero. This is explained by the fact that some possible deviations are positive, others are negative, and as a result of their mutual cancellation, zero is obtained.

Dispersion (scattering) Discrete random variable is called the mathematical expectation of the squared deviation of the random variable from its mathematical expectation.

In practice, this method of calculating the variance is inconvenient, because leads to cumbersome calculations for a large number of values ​​of a random variable.

Therefore, another method is used.

Theorem. The variance is equal to the difference between the mathematical expectation of the square of the random variable X and the square of its mathematical expectation.

Proof. Taking into account the fact that the mathematical expectation M (X) and the square of the mathematical expectation M 2 (X) are constant values, we can write:

Example. Find the variance of a discrete random variable given by the distribution law.

Decision: .

6.1.4 Dispersion properties

1. The dispersion of a constant value is zero. .

2. A constant factor can be taken out of the dispersion sign by squaring it. .

3. The variance of the sum of two independent random variables is equal to the sum of the variances of these variables. .

4. The variance of the difference of two independent random variables is equal to the sum of the variances of these variables. .

Theorem. The variance of the number of occurrences of event A in n independent trials, in each of which the probability p of the occurrence of the event is constant, is equal to the product of the number of trials and the probabilities of occurrence and non-occurrence of the event in each trial.

Example: Find the variance of DSV X - the number of occurrences of event A in 2 independent trials, if the probability of occurrence of the event in these trials is the same and it is known that M(X) = 1.2.

We apply the theorem from Section 6.1.2:

M(X) = np

M(X) = 1,2; n= 2. Find p:

1,2 = 2∙p

p = 1,2/2

q = 1 – p = 1 – 0,6 = 0,4

Let's find the dispersion by the formula:

D(X) = 2∙0,6∙0,4 = 0,48

6.1.5 Standard deviation of a discrete random variable

Standard deviation random variable X is called the square root of the variance.

(25)

Theorem. The standard deviation of the sum of a finite number of mutually independent random variables is equal to the square root of the sum of the squared standard deviations of these variables.

6.1.6 Mode and median of a discrete random variable

Fashion M o DSV the most probable value of a random variable is called (i.e. the value that has the highest probability)

Median M e DSW is the value of a random variable that divides the distribution series in half. If the number of values ​​of the random variable is even, then the median is found as the arithmetic mean of the two mean values.

Example: Find Mode and Median of DSW X:

Me = = 5,5

Working process

1. Get acquainted with the theoretical part of this work (lectures, textbook).

2. Complete the task according to your choice.

3. Compile a report on the work.

4. Protect your work.

2. The purpose of the work.

3. Progress of work.

4. Decision of your option.

6.4 Variants of tasks for independent work

Option number 1

1. Find the mathematical expectation, variance, standard deviation, mode and median of the DSV X given by the distribution law.

2. Find the mathematical expectation of a random variable Z, if the mathematical expectations of X and Y are known: M(X)=6, M(Y)=4, Z=5X+3Y.

3. Find the variance of DSV X - the number of occurrences of event A in two independent trials, if the probabilities of occurrence of events in these trials are the same and it is known that M (X) = 1.

4. A list of possible values ​​of a discrete random variable is given X: x 1 = 1, x2 = 2, x 3= 5, and the mathematical expectations of this quantity and its square are also known: , . Find the probabilities , , , corresponding to the possible values ​​, , and draw up the distribution law of the DSW.

Option number 2

2. Find the mathematical expectation of a random variable Z, if the mathematical expectations of X and Y are known: M(X)=5, M(Y)=8, Z=6X+2Y.

3. Find the variance of DSV X - the number of occurrences of event A in three independent trials, if the probabilities of occurrence of events in these trials are the same and it is known that M (X) = 0.9.

4. A list of possible values ​​of a discrete random variable X is given: x 1 = 1, x2 = 2, x 3 = 4, x4= 10, and the mathematical expectations of this quantity and its square are also known: , . Find the probabilities , , , corresponding to the possible values ​​, , and draw up the distribution law of the DSW.

Option number 3

1. Find the mathematical expectation, variance and standard deviation of the DSV X given by the distribution law.

2. Find the mathematical expectation of a random variable Z, if the mathematical expectations of X and Y are known: M(X)=3, M(Y)=4, Z=4X+2Y.

3. Find the variance of DSV X - the number of occurrences of event A in four independent trials, if the probabilities of occurrence of events in these trials are the same and it is known that M (x) = 1.2.

1. The mathematical expectation of a constant value is equal to the constant itself M(S)=S .
2. A constant factor can be taken out of the expectation sign: M(CX)=CM(X)
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).
4. The mathematical expectation of the sum of two random variables is equal to the sum of the mathematical expectations of the terms: M(X+Y)=M(X)+M(Y).

Theorem. The mathematical expectation M(x) of the number of occurrences of events A in n independent trials is equal to the product of these trials by the probability of occurrence of events in each trial: M(x) = np.

Let be X is a random variable and M(X) is its mathematical expectation. Consider as a new random variable the difference X - M(X).

Deviation is the difference between a random variable and its mathematical expectation.

The deviation has the following distribution law:

Solution: Find the mathematical expectation:
2 =(1-2.3) 2 =1.69
2 =(2-2.3) 2 =0.09
2 =(5-2.3) 2 =7.29

Let's write the distribution law of the squared deviation:

Solution: Find the expectation M(x): M(x)=2 0.1+3 0.6+5 0.3=3.5

Let's write the distribution law of the random variable X 2

Let's find the mathematical expectation M(x2):M(x2) = 4 0.1+9 0.6+25 0.3=13.5

The desired dispersion D (x) \u003d M (x 2) - 2 \u003d 13.3- (3.5) 2 \u003d 1.05

Dispersion properties:

1. Dispersion of a constant value With equals zero: D(C)=0
2. A constant factor can be taken out of the dispersion sign by squaring it. D(Cx)=C 2 D(x)
3. The variance of the sum of independent random variables is equal to the sum of the variances of these variables. D(X 1 +X 2 +...+X n)=D(X 1)+D(X 2)+...+D(X n)
4. The variance of the binomial distribution is equal to the product of the number of trials and the probability of occurrence and non-occurrence of an event in one trial D(X)=npq

To estimate the dispersion of possible values ​​of a random variable around its mean value, in addition to the variance, some other characteristics also serve. Among them is the standard deviation.

The standard deviation of a random variable X called the square root of the variance:

σ(X) = √D(X) (4)

Example. The random variable X is given by the distribution law

Find the standard deviation σ(x)

Solution: Find the mathematical expectation X: M(x)=2 0.1+3 0.4+10 0.5=6.4
Let's find the mathematical expectation of X 2: M(x 2)=2 2 0.1+3 2 0.4+10 2 0.5=54
Find the dispersion: D(x)=M(x 2)=M(x 2)- 2 =54-6.4 2 =13.04
Desired standard deviation σ(X)=√D(X)=√13.04≈3.61

Theorem. The standard deviation of the sum of a finite number of mutually independent random variables is equal to the square root of the sum of the squared standard deviations of these variables:

Example. There are 3 books on mathematics and 3 on physics on a shelf of 6 books. Three books are chosen at random. Find the law of distribution of the number of books in mathematics among the selected books. Find the mathematical expectation and variance of this random variable.

D (X) \u003d M (X 2) - M (X) 2 \u003d 2.7 - 1.5 2 \u003d 0.45