Mathematical expectation is the probability distribution of a random variable. random variables
As already known, the distribution law completely characterizes a 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 of a random variable.
Mathematical expectation is one of the important numerical characteristics.
The mathematical expectation is approximately equal to the average value of a random variable.
Mathematical expectation of a discrete random variable is the sum of the products of all its possible values and their probabilities.
If a random variable is characterized by a finite distribution series:
| X | x 1 | x 2 | x 3 | … | x n |
| R | p 1 | p 2 | p 3 | … | r p |
then the mathematical expectation M(X) is determined by the formula:
The mathematical expectation of a continuous random variable is determined by the equality:
where is the probability density of the random variable X.
Example 4.7. Find the mathematical expectation of the number of points that fall out when a dice is thrown.
Solution:
Random value X takes the values 1, 2, 3, 4, 5, 6. Let's make the law of its distribution:
| X | ||||||
| R |
Then the mathematical expectation is:
Properties of mathematical expectation:
1. The mathematical expectation of a constant value is equal to the constant itself:
M(S)=S.
2. The 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).
Example 4.8. Independent random variables X And Y are given by the following distribution laws:
| X | Y | ||||||
| R | 0,6 | 0,1 | 0,3 | R | 0,8 | 0,2 |
Find the mathematical expectation of a random variable XY.
Solution.
Let's find the mathematical expectations of each of these quantities:
random variables X And Y independent, so the desired mathematical expectation:
M(XY) = M(X)M(Y)=
Consequence. The mathematical expectation of the product of several mutually independent random variables is equal to the product of their mathematical expectations.
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).
Consequence. The mathematical expectation of the sum of several random variables is equal to the sum of the mathematical expectations of the terms.
Example 4.9. 3 shots are fired with probabilities of hitting the target equal to p 1 = 0,4; p2= 0.3 and p 3= 0.6. Find the mathematical expectation of the total number of hits.
Solution.
The number of hits on the first shot is a random variable X 1, which can take only two values: 1 (hit) with probability p 1= 0.4 and 0 (miss) with probability q 1 = 1 – 0,4 = 0,6.
The mathematical expectation of the number of hits in the first shot is equal to the probability of hitting:
Similarly, we find the mathematical expectations of the number of hits in the second and third shots:
M(X 2)= 0.3 and M (X 3) \u003d 0,6.
The total number of hits is also a random variable consisting of the sum of hits in each of the three shots:
X \u003d X 1 + X 2 + X 3.
The desired mathematical expectation X we find by the theorem of mathematical, the expectation of the sum.
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 , That:
If F(x)- distribution function X, then the mathematical expectation is representable integralLebesgue - Stieltjes (or Riemann - Stieltjes):
while the integrability X In terms of ( * ) 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), That
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 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.
Characteristics of DSW and their properties. Mathematical expectation, variance, standard deviation
The distribution law fully characterizes the random variable. However, when it is impossible to find the distribution law, or this is not required, one can limit oneself to finding values, called numerical characteristics of a random variable. These values determine some average value around which the values of a random variable are grouped, and the degree of their dispersion around this average value.
mathematical expectation A discrete random variable is the sum of the products of all possible values of a random variable and their probabilities.
The mathematical expectation exists if the series on the right side of the equality converges absolutely.
From the point of view of probability, we can say that the mathematical expectation is approximately equal to the arithmetic mean of the observed values of the random variable.
Example. The law of distribution of a discrete random variable is known. Find the mathematical expectation.
| X | ||||
| p | 0.2 | 0.3 | 0.1 | 0.4 |
Solution:
9.2 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.
Let n independent trials be performed, the probability of occurrence of event A in which is equal to p.
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.
Example. Find the mathematical expectation of a random variable Z if the mathematical expectations of X and Y are known: M(X)=3, M(Y)=2, Z=2X+3Y.
Solution:
9.3 Dispersion of a discrete random variable
However, the 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.
| X | ||||
| X 2 | ||||
| R | 0.2 | 0.3 | 0.1 | 0.4 |
Solution: .
9.4 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 probability of occurrence and non-occurrence of the event in each trial.
9.5 Standard deviation of a discrete random variable
Standard deviation random variable X is called the square root of the variance.
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.
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
- 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
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 |
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
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
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
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
Thus, 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
Thus, 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
