Probability theory formula of mathematical expectation. Expectation Formula

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.

Example for 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 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 value 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 probabilities p1, p2, p3 corresponding to 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 distribution function contains full information about a random variable. In practice, the allocation function cannot always be established; sometimes such exhaustive knowledge is not required. Partial information about a random variable is given by numerical characteristics, which, depending on the type of information, are divided into the following groups.
1. Characteristics of the position of a random variable on the numerical axis (mode Mo, median Me, expected value M(X)).
2. Characteristics of the spread of a random variable around the mean value (dispersion D(X), standard deviation σ( X)).
3. Characteristics of the curve shape y = φ( x) (asymmetry As, kurtosis Ex).
Let's take a closer look at each of these characteristics.
Expected value random variable X indicates some average value around which all possible values ​​are grouped X. For a discrete random variable that can take only a finite number of possible values, the mathematical expectation is the sum of the products of all possible values ​​of the random variable and the probability of these values:
. (2.4)
For a continuous random variable X, which has a given distribution density φ( x) the mathematical expectation is the following integral:
. (2.5)
Here it is assumed that improper integral converges absolutely, i.e. exists.
Properties of mathematical expectation:
1. M(S) = C, where FROM = const;
2. M(C∙X) = C∙M(X);
3. M(X ± Y) = M(X) ± M(Y), where X and Y– any random variables;
4. M(X∙Y)=M(X)∙M(Y), where X and Y are independent random variables.
Two random variables are called independent , if the distribution law of one of them does not depend on what possible values ​​the other value has taken.
Fashion discrete random variable, denoted Mo, its most probable value is called (Fig. 2.3), and the mode of a continuous random variable is the value at which the probability density is maximum (Fig. 2.4).



Rice. 2.3 Fig. 2.4
median continuous random variable X its value Me is called such, for which it is equally probable whether the random variable will turn out to be less or more Me, i.e.
P(X < Me) = P(X > Me)
From the definition of the median, it follows that P(X<Me) = 0.5, i.e. F (Me) = 0.5. Geometrically, the median can be interpreted as the abscissa, in which the ordinate φ( x) bisects the area bounded by the distribution curve (Fig. 2.5). In the case of a symmetrical distribution, the median coincides with the mode and the mathematical expectation (Fig. 2.6).

Rice. 2.5 Fig. 2.6

Dispersion.

Variance of a random variable- a measure of the spread of a given random variable, that is, its deviation from the mathematical expectation. Denoted D[X] in Russian literature and (eng. variance) in foreign countries. In statistics, the designation or is often used. The square root of the variance, equal to , is called the standard deviation, standard deviation, or standard spread. The standard deviation is measured in the same units as the random variable itself, and the variance is measured in the squares of that unit.

It follows from Chebyshev's inequality that a random variable moves away from its mathematical expectation by more than k standard deviations with probability less than 1/ k². So, for example, in at least 75% of cases, the random variable is removed from its mean by no more than two standard deviations, and in about 89% - by no more than three.

dispersion random variable is called the mathematical expectation of the square of its deviation from the mathematical expectation
D(X) = M(X –M(X)) 2 .
Variance of a random variable X it is convenient to calculate by the formula:
a) for a discrete quantity
; (2.6)
b) for a continuous random variable
j( X)d x – 2 . (2.7)
The dispersion has the following properties:
1. D(C) = 0, where FROM = const;
2. D(C× X) = C 2 ∙ D(X);
3. D(X± Y) = D(X) + D(Y), if X and Y independent random variables.
Standard deviation random variable X is called the arithmetic root of the variance, i.e.
σ( X) = .
Note that the dimension σ( X) coincides with the dimension of the random variable itself X, so the standard deviation is more convenient for scattering characterization.
A generalization of the main numerical characteristics of random variables is the concept of moments of a random variable.
The initial moment of the kth order α k random variable X is called the mathematical expectation of the quantity X k, i.e. α k = M(X k).
The initial moment of the first order is the mathematical expectation of the random variable.
The central moment of the kth order μ k random variable X is called the mathematical expectation of the quantity ( X–M(X))k, i.e. μ k = M(X–M(X))k.
The central moment of the second order is the variance of the random variable.
For a discrete random variable, the initial moment is expressed by the sum α k= , and the central one is the sum μ k = where p i = p(X=x i). For the initial and central moments of a continuous random variable, the following equalities can be obtained:
α k = ,  μ k = ,
where φ( x) is the distribution density of the random variable X.
Value As= μ 3 / σ 3 is called asymmetry coefficient .
If the asymmetry coefficient is negative, then this indicates a large influence on the value of m 3 negative deviations. In this case, the distribution curve (Fig. 2.7) is more flat to the left of M(X). If the coefficient As is positive, which means that the influence of positive deviations prevails, then the distribution curve (Fig. 2.7) is flatter on the right. In practice, the sign of the asymmetry is determined by the location of the distribution curve relative to the mode (maximum point of the differential function).


Rice. 2.7
kurtosis Ek is called the quantity
Ek\u003d μ 4 / σ 4 - 3.

Question 24: Correlation

Correlation (correlation dependence) - statistical relationship of two or more random variables (or variables that can be considered as such with some acceptable degree of accuracy). In this case, changes in the values ​​of one or more of these quantities are accompanied by a systematic change in the values ​​of another or other quantities. The mathematical measure of the correlation of two random variables is correlation relation, or correlation coefficient (or ) . If a change in one random variable does not lead to a regular change in another random variable, but leads to a change in another statistical characteristic of this random variable, then such a relationship is not considered a correlation, although it is statistical.

For the first time, the term “correlation” was introduced into scientific circulation by the French paleontologist Georges Cuvier in the 18th century. He developed the "law of correlation" of parts and organs of living beings, with the help of which it is possible to restore the appearance of a fossil animal, having at its disposal only a part of its remains. In statistics, the word "correlation" was first used by the English biologist and statistician Francis Galton at the end of the 19th century.

Some types of correlation coefficients can be positive or negative (it is also possible that there is no statistical relationship - for example, for independent random variables). If it is assumed that a strict order relation is given on the values ​​of the variables, then negative correlation- correlation, in which an increase in one variable is associated with a decrease in another variable, while the correlation coefficient can be negative; positive correlation in such conditions, a correlation in which an increase in one variable is associated with an increase in another variable, while the correlation coefficient can be positive.

Average values ​​of random variables

Let's pretend that X is a discrete random variable, which, as a result of the experiment, took the values x 1 , x 2 ,…, x n with probabilities p 1 , p 2 ,…, p n, . Then the mean value or mathematical expectation of the quantity X called the sum , i.e. the weighted average value of X, where the weights are the probabilities pi.

Example. Determine the average value of the control error e, if, on the basis of a large number of experiments, it is established that the error probability p i is equal to:

1. M[e] = 0.1×0.2 + 0.15×0.2 + 0.2×0.3 + 0.25×0.15 + 0.3×0.15 =

If g( X) is a function X(and the probability that X = x i is equal to pi), then the mean value of the function is defined as

Let's pretend that X is a random variable with a continuous distribution and is characterized by a probability density j( x). Then the probability that X is between x and x+ D X:

The value of X then approximately takes the value x. In the limit at D x® 0, we can assume that the increment D x numerically equal to the differential d x.

By replacing D x=d X, we obtain the exact formula for calculating the average value X :

Similarly for g( X):

As a rule, it is not enough to know only the mean value (mathematical expectation) of a random variable. To estimate the measure of randomness of a quantity (to estimate the spread of specific values X regarding mathematical expectation M[X]) introduces the concept of dispersion of a random variable. Dispersion - the average value of the squared deviation of each specific value of X from the mathematical expectation. The greater the dispersion, the greater the randomness of the spread of the value from the mathematical expectation. If the random variable is discrete, then

For a continuous random variable, the variance can be written similarly:

The dispersion well describes the spread of the value, but there is one drawback: the dimension does not correspond to the dimension X. To get rid of this shortcoming, often in specific applications they consider not , but a positive value, which is called standard deviation.

1.3.2.1. Expectation Properties

1. The mathematical expectation of a non-random variable is equal to this value itself M[C] = C.

2. Non-random multiplier FROM can be taken out of the mathematical expectation sign M[CX] = CM[X].

3. The mathematical expectation of the sum of random variables is equal to the sum of the mathematical expectations of these random variables.

4. The mathematical expectation of the product of independent random variables is equal to the product of the mathematical expectations of these variables (the condition for the independence of random variables).

1.3.2.2. Dispersion Properties

1. Dispersion of a non-random variable FROM equals zero: D[C]=0.

2. Dispersion of the product of a non-random factor FROM by a random variable is equal to the product FROM 2 on the variance of a random variable.

3. Dispersion of the sum of independent random variables X 1 and X 2 is equal to the sum of the dispersions of the terms

1.3.3. Moments of a random variable

Let X is a continuous random variable. If n is a positive integer and the function x n is integrable on the interval (–¥; +¥), then the average value

n = 0, 1,…, n

called initial moment order n random variable X.

Obviously, the moment of zero order

,

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 .

In the simplest way, the 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 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 by 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 (mean 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

1. The mathematical expectation of a constant value is equal to itself: M[C]=C , C is a constant;
2. M=C M[X]
3. The mathematical expectation of the sum of random variables is equal to the sum of their mathematical expectations: M=M[X]+M[Y]
4. 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

1. The dispersion of a constant value is equal to zero: D(c)=0.
2. The constant factor can be taken out from under the dispersion sign by squaring it: D(k*X)= k 2 D(X).
3. 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).
4. If random variables X and Y are dependent: D(X+Y)=DX+DY+2(X-M[X])(Y-M[Y])
5. 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

1. Multiply the pairs one by one: x i by p i .
2. 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

Example #1.

Example #2. A discrete random variable has the following distribution series:

Solution. The value a is found from the relationship: 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