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 random variable in short 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 original 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 example, if X- random variable with values in and f(x)- unambiguous Borelfunction X , then:
If a 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 next most important property of a random variable after the mathematical expectation is its variance, defined as the mean square of the deviation from the mean:
If denoted by then, the variance VX will be the expected value. This is a characteristic of the "scatter" of the X distribution.
As a simple example calculating variance, suppose we've just been made an offer we can't refuse: someone gave us two certificates to participate in the same lottery. The organizers of the lottery sell 100 tickets every week, participating in a separate draw. One of these tickets is selected in a draw through a uniform random process - each ticket has an equal chance of being selected - and the owner of that lucky ticket receives one hundred million dollars. The remaining 99 lottery ticket holders win nothing.
We can use the gift in two ways: either buy two tickets in the same lottery, or one ticket each to participate in two different lotteries. What is the best strategy? Let's try to analyze. To do this, we denote by random variables representing the size of our winnings on the first and second tickets. The expected value in millions is
and the same is true for the expected values are additive, so our average total payoff will be
regardless of the strategy adopted.
However, the two strategies appear to be different. Let's go beyond the expected values and study the entire probability distribution
If we buy two tickets in the same lottery, we have a 98% chance of winning nothing and a 2% chance of winning 100 million. If we buy tickets for different draws, then the numbers will be as follows: 98.01% - the chance of not winning anything, which is somewhat higher than before; 0.01% - a chance to win 200 million, also a little more than it was before; and the chance of winning 100 million is now 1.98%. Thus, in the second case, the distribution of magnitude is somewhat more scattered; the average, $100 million, is somewhat less likely, while the extremes are more likely.
It is this concept of the scatter of a random variable that is intended to reflect the variance. We measure the spread through the square of the deviation of a random variable from its mathematical expectation. Thus, in case 1, the variance will be
in case 2, the variance is
As we expected, the latter value is somewhat larger, since the distribution in case 2 is somewhat more scattered.
When we work with variances, everything is squared, so the result can be quite large numbers. (The multiplier is one trillion, that should be impressive
even players accustomed to high stakes.) Square root from dispersion. The resulting number is called the standard deviation and is usually denoted Greek letter a:
The standard deviations for our two lottery strategies are . In some ways, the second option is about $71,247 riskier.
How does variance help in choosing a strategy? It's not clear. A strategy with a larger variance is riskier; but what is better for our wallet - risk or safe play? Let us have the opportunity to buy not two tickets, but all one hundred. Then we could guarantee a win in one lottery (and the variance would be zero); or you could play in a hundred different draws, getting nothing with probability, but having a non-zero chance of winning up to dollars. Choosing one of these alternatives is beyond the scope of this book; all we can do here is explain how to make the calculations.
In fact, there is an easier way to calculate the variance than using definition (8.13) directly. (There is every reason to suspect some hidden mathematics here; otherwise, why would the variance in the lottery examples turn out to be an integer multiple) We have
because is a constant; Consequently,
"Dispersion is the mean of the square minus the square of the mean"
For example, in the lottery problem, the mean is or Subtraction (of the square of the mean) gives results that we have already obtained earlier in a more difficult way.
There is, however, an even simpler formula that applies when we calculate for independent X and Y. We have
since, as we know, for independent random variables Hence,
"The variance of the sum of independent random variables is equal to the sum of their variances" So, for example, the variance of the amount that can be won on one lottery ticket is equal to
Therefore, the variance of the total winnings for two lottery tickets in two different (independent) lotteries will be The corresponding value of the variance for independent lottery tickets will be
The variance of the sum of points rolled on two dice can be obtained using the same formula, since there is a sum of two independent random variables. We have
for the correct cube; therefore, in the case of a displaced center of mass
therefore, if the center of mass of both cubes is displaced. Note that in the latter case, the variance is larger, although it takes an average of 7 more often than in the case of regular dice. If our goal is to roll more lucky sevens, then the variance is not best indicator success.
Okay, we have established how to calculate the variance. But we have not yet given an answer to the question of why it is necessary to calculate the variance. Everyone does it, but why? The main reason is the Chebyshev inequality which establishes an important property of the variance:
(This inequality differs from Chebyshev's inequalities for sums, which we encountered in Chapter 2.) Qualitatively, (8.17) states that a random variable X rarely takes values far from its mean if its variance VX is small. Proof
action is extraordinarily simple. Really,
division by completes the proof.
If we denote the mathematical expectation through a and the standard deviation - through a and replace in (8.17) with then the condition turns into therefore, we get from (8.17)
Thus, X will lie within - times the standard deviation of its mean except in cases where the probability does not exceed Random value will lie within 2a of at least 75% of the trials; ranging from to - at least for 99%. These are cases of Chebyshev's inequality.
If you roll a couple of dice times, then total amount points in all throws almost always, for large ones it will be close to The reason for this is as follows: the variance of independent throws will be Dispersion in means the standard deviation of the total
Therefore, from the Chebyshev inequality, we obtain that the sum of points will lie between
for at least 99% of all rolls of the correct dice. For example, the total of a million tosses with a probability of more than 99% will be between 6.976 million and 7.024 million.
In the general case, let X be any random variable on the probability space P that has a finite mathematical expectation and a finite standard deviation a. Then we can introduce into consideration the probability space Пп, whose elementary events are -sequences where each , and the probability is defined as
If we now define random variables by the formula
then the value
will be the sum of independent random variables, which corresponds to the process of summation of independent realizations of the quantity X on P. The mathematical expectation will be equal to and the standard deviation - ; therefore, the mean value of realizations,
will lie in the range from to at least 99% of the time period. In other words, if one chooses a large enough value, then the arithmetic mean of independent trials will almost always be very close to the expected value. big numbers; but the simple corollary of Chebyshev's inequality, which we just derived, suffices for us.)
Sometimes we do not know the characteristics of the probability space, but we need to estimate the mathematical expectation of a random variable X by repeated observations of its value. (For example, we might want the mean January midday temperature in San Francisco; or we might want to know the life expectancy on which to base our calculations. insurance agents.) If we have independent empirical observations at our disposal, then we can assume that the true mathematical expectation is approximately equal to
You can also estimate the variance using the formula
Looking at this formula, one might think that there is a typographical error in it; it would seem that there should be as in (8.19), since the true value of the variance is determined in (8.15) through the expected values. However, the substitution here for allows us to obtain best estimate, since definition (8.20) implies that
Here is the proof:
(In this calculation, we rely on the independence of observations when we replace by )
In practice, to evaluate the results of an experiment with a random variable X, one usually calculates the empirical mean and the empirical standard deviation and then writes the answer in the form Here, for example, are the results of throwing a pair of dice, supposedly correct.
Random variables, in addition to distribution laws, can also be described numerical characteristics .
mathematical expectation M (x) of a random variable is called its average value.
The mathematical expectation of a discrete random variable is calculated by the formula
where – values of a random variable, p i- their probabilities.
Consider the properties of mathematical expectation:
1. The mathematical expectation of a constant is equal to the constant itself
2. If a random variable is multiplied by a certain number k, then the mathematical expectation will be multiplied by the same number
M (kx) = kM (x)
3. The mathematical expectation of the sum of random variables is equal to the sum of their mathematical expectations
M (x 1 + x 2 + ... + x n) \u003d M (x 1) + M (x 2) + ... + M (x n)
4. M (x 1 - x 2) \u003d M (x 1) - M (x 2)
5. For independent random variables x 1 , x 2 , … x n the mathematical expectation of the product is equal to the product of their mathematical expectations
M (x 1, x 2, ... x n) \u003d M (x 1) M (x 2) ... M (x n)
6. M (x - M (x)) \u003d M (x) - M (M (x)) \u003d M (x) - M (x) \u003d 0
Let's calculate the mathematical expectation for the random variable from Example 11.
M(x) == 
.
Example 12. Let the random variables x 1 , x 2 be given by the distribution laws, respectively:
x 1 Table 2
x 2 Table 3
Calculate M (x 1) and M (x 2)
M (x 1) \u003d (- 0.1) 0.1 + (- 0.01) 0.2 + 0 0.4 + 0.01 0.2 + 0.1 0.1 \u003d 0
M (x 2) \u003d (- 20) 0.3 + (- 10) 0.1 + 0 0.2 + 10 0.1 + 20 0.3 \u003d 0
The mathematical expectations of both random variables are the same - they are equal to zero. However, their distribution is different. If the values of x 1 differ little from their mathematical expectation, then the values of x 2 differ to a large extent from their mathematical expectation, and the probabilities of such deviations are not small. These examples show that it is impossible to determine from the average value what deviations from it take place both up and down. So with the same average The annual precipitation in two localities cannot be said to be equally favorable for agricultural work. Similarly, in terms of average wages it is not possible to judge specific gravity high and low paid workers. Therefore, a numerical characteristic is introduced - dispersion D(x) , which characterizes the degree of deviation of a random variable from its mean value:
D (x) = M (x - M (x)) 2 . (2)
Dispersion is the mathematical expectation of the squared deviation of a random variable from the mathematical expectation. For a discrete random variable, the variance is calculated by the formula:
D(x)= 
= 
(3)
It follows from the definition of variance that D (x) 0.
Dispersion properties:
1. Dispersion of the constant is zero
2. If a random variable is multiplied by some number k, then the variance is multiplied by the square of this number
D (kx) = k 2 D (x)
3. D (x) \u003d M (x 2) - M 2 (x)
4. For pairwise independent random variables x 1 , x 2 , … x n the variance of the sum is equal to the sum of the variances.
D (x 1 + x 2 + ... + x n) = D (x 1) + D (x 2) + ... + D (x n)
Let's calculate the variance for the random variable from Example 11.
Mathematical expectation M (x) = 1. Therefore, according to the formula (3) we have:
D (x) = (0 – 1) 2 1/4 + (1 – 1) 2 1/2 + (2 – 1) 2 1/4 =1 1/4 +1 1/4= 1/2
Note that it is easier to calculate the variance if we use property 3:
D (x) \u003d M (x 2) - M 2 (x).
Let's calculate the variances for random variables x 1 , x 2 from Example 12 using this formula. The mathematical expectations of both random variables are equal to zero.
D (x 1) \u003d 0.01 0.1 + 0.0001 0.2 + 0.0001 0.2 + 0.01 0.1 \u003d 0.001 + 0.00002 + 0.00002 + 0.001 \u003d 0.00204
D (x 2) \u003d (-20) 2 0.3 + (-10) 2 0.1 + 10 2 0.1 + 20 2 0.3 \u003d 240 +20 \u003d 260
The closer the dispersion value is to zero, the smaller the spread of the random variable relative to the mean value.
The value is called standard deviation. Random fashion x discrete type Md is the value of the random variable, which corresponds to the highest probability.
Random fashion x continuous type Md, is a real number defined as the maximum point of the probability distribution density f(x).
Median of a random variable x continuous type Mn is a real number that satisfies the equation
The mathematical expectation is the average value of a random variable.The mathematical expectation of a discrete random variable is the sum of the products of all its possible values and their probabilities:
Example.
X -4 6 10
p 0.2 0.3 0.5
Solution: The mathematical expectation is equal to the sum of the products of all possible values of X and their probabilities:
M (X) \u003d 4 * 0.2 + 6 * 0.3 + 10 * 0.5 \u003d 6.
To calculate the mathematical expectation, it is convenient to carry out calculations in Excel (especially when there is a lot of data), we suggest using ready template ().
An example for an independent solution (you can use a calculator).
Find the mathematical expectation of a discrete random variable X given by the distribution law:
X 0.21 0.54 0.61
p 0.1 0.5 0.4
Mathematical expectation has the following properties.
Property 1. The mathematical expectation of a constant value is equal to the constant itself: М(С)=С.
Property 2. A constant factor can be taken out of the expectation sign: М(СХ)=СМ(Х).
Property 3. The mathematical expectation of the product of mutually independent random variables is equal to the product of the mathematical expectations of the factors: M (X1X2 ... Xp) \u003d M (X1) M (X2) *. ..*M(Xn)
Property 4. The mathematical expectation of the sum of random variables is equal to the sum of the mathematical expectations of the terms: М(Хг + Х2+...+Хn) = М(Хг)+М(Х2)+…+М(Хn).
Problem 189. Find the mathematical expectation of a random variable Z if the mathematical expectations X and Y are known: Z = X+2Y, M(X) = 5, M(Y) = 3;
Solution: Using the properties of the mathematical expectation (the mathematical expectation of the sum is equal to the sum of the mathematical expectations of the terms; the constant factor can be taken out of the mathematical expectation sign), we get M(Z)=M(X + 2Y)=M(X) + M(2Y)=M (X) + 2M(Y)= 5 + 2*3 = 11.
190. Using the properties of mathematical expectation, prove that: a) M(X - Y) = M(X)-M (Y); b) the mathematical expectation of the deviation X-M(X) is zero.
191. Discrete random variable X takes three possible values: x1= 4 With probability p1 = 0.5; x3 = 6 With probability P2 = 0.3 and x3 with probability p3. Find: x3 and p3, knowing that M(X)=8.
192. A list of possible values of a discrete random variable X is given: x1 \u003d -1, x2 \u003d 0, x3 \u003d 1, the mathematical expectations of this quantity and its square are also known: M (X) \u003d 0.1, M (X ^ 2) \u003d 0 ,9. Find the probabilities p1, p2, p3 corresponding possible values xi
194. A batch of 10 parts contains three non-standard parts. Two items were selected at random. Find the mathematical expectation of a discrete random variable X - the number of non-standard parts among two selected ones.
196. Find the mathematical expectation of a discrete random variable X-number of such throws of five dice, in each of which one point will appear on two bones, if total number throws equal to twenty.
The mathematical expectation of the binomial distribution is equal to the product of the number of trials and the probability of an event occurring in one trial:
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, with the help of simple calculations, you can find the average arithmetic value dropped points.
As well as dropping any of the range values, this value will be random.
And if you increase the number of throws several times? At large quantities throws, the arithmetic mean value of the points will approach a specific number, which in the theory of probability 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 fields 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 know average height dwarfs in the tale of 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 can be calculated possible number production of defective parts.
Mathematical expectation also turns out to be indispensable during the statistical processing of the data obtained in the course of scientific research results. 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 reaching positive results no less 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 by positive expectation and commit a large number transactions over a short period of time.
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%. The calculation of the mathematical expectation for the transaction gives a value of 625 dollars.
