The random variable x is given by the probability distribution function. A2 - random variable X has taken the value x2. Distribution function of a continuous random variable and probability density

Examples of solving problems on the topic " random variables».

Task 1 . There are 100 tickets issued in the lottery. One win of 50 USD was played. and ten wins of $10 each. Find the law of distribution of the value X - the cost of a possible gain.

Solution. Possible values ​​of X: x 1 = 0; x 2 = 10 and x 3 = 50. Since there are 89 “empty” tickets, then p 1 = 0.89, the probability of winning is 10 c.u. (10 tickets) – p 2 = 0.10 and for a win of 50 c.u. –p 3 = 0.01. Thus:

	0,89	0,10	0,01

Easy to control: .

Task 2. The probability that the buyer has familiarized himself with the advertisement of the product in advance is 0.6 (p = 0.6). Selective quality control of advertising is carried out by polling buyers before the first one who has studied the advertisement in advance. Make a series of distribution of the number of interviewed buyers.

Solution. According to the condition of the problem p = 0.6. From: q=1 -p = 0.4. Substituting these values, we get: and construct a distribution series:

pi		0,24

Task 3. A computer consists of three independently operating elements: a system unit, a monitor, and a keyboard. With a single sharp increase in voltage, the probability of failure of each element is 0.1. Based on the Bernoulli distribution, draw up the distribution law for the number of failed elements during a power surge in the network.

Solution. Consider Bernoulli distribution(or binomial): the probability that in n tests, event A will appear exactly k once: , or:

	q n					p n

IN let's get back to the task.

Possible values ​​of X (number of failures):

x 0 =0 - none of the elements failed;

x 1 =1 - failure of one element;

x 2 =2 - failure of two elements;

x 3 =3 - failure of all elements.

Since, by condition, p = 0.1, then q = 1 – p = 0.9. Using the Bernoulli formula, we get

, ,

, .

Control: .

Therefore, the desired distribution law:

	0,729	0,243	0,027	0,001

Task 4. Produced 5000 rounds. The probability that one cartridge is defective . What is the probability that there will be exactly 3 defective cartridges in the entire batch?

Solution. Applicable Poisson distribution: this distribution is used to determine the probability that, given a very large

number of trials (mass trials), in each of which the probability of event A is very small, event A will occur k times: , Where .

Here n \u003d 5000, p \u003d 0.0002, k \u003d 3. We find , then the desired probability: .

Task 5. When firing before the first hit with the probability of hitting p = 0.6 for a shot, you need to find the probability that the hit will occur on the third shot.

Solution. Let us apply the geometric distribution: let independent trials be performed, in each of which the event A has a probability of occurrence p (and non-occurrence q = 1 - p). Trials end as soon as event A occurs.

Under such conditions, the probability that event A will occur on the kth test is determined by the formula: . Here p = 0.6; q \u003d 1 - 0.6 \u003d 0.4; k \u003d 3. Therefore, .

Task 6. Let the law of distribution of a random variable X be given:

Find the mathematical expectation.

Solution. .

Note that the probabilistic meaning mathematical expectation is the mean value of the random variable.

Task 7. Find the variance of a random variable X with the following distribution law:

Solution. Here .

The law of distribution of the square of X 2 :

X 2

Required variance: .

Dispersion characterizes the degree of deviation (scattering) of a random variable from its mathematical expectation.

Task 8. Let the random variable be given by the distribution:

			10m

Find its numerical characteristics.

Solution: m, m 2 ,

M 2 , m.

About a random variable X, one can say either - its mathematical expectation is 6.4 m with a variance of 13.04 m 2 , or - its mathematical expectation is 6.4 m with a deviation of m. The second formulation is obviously clearer.

Task 9. Random value X given by the distribution function:
.

Find the probability that, as a result of the test, the value X will take on a value contained in the interval .

Solution. The probability that X will take a value from a given interval is equal to the increment of the integral function in this interval, i.e. . In our case and , therefore

.

Task 10. Discrete random variable X given by the distribution law:

Find distribution function F(x ) and build its graph.

Solution. Because distribution function,

For , That

at ;

at ;

at ;

at ;

Relevant chart:

Task 11. Continuous random variable X given differential function distributions: .

Find the probability of hitting X to interval

Solution. Note that this is a special case of the exponential distribution law.

Let's use the formula: .

Task 12. Find the numerical characteristics of a discrete random variable X given by the distribution law:

	–5
	X 2 : x2 . , Where is the Laplace function. The values ​​of this function are found using a table. In our case: . According to the table we find:, therefore:

Random variable A quantity is called that, as a result of tests carried out under the same conditions, takes on different, generally speaking, values, depending on random factors that are not taken into account. Examples of random variables: the number of points dropped on a dice, the number of defective items in a batch, the deviation of the point of impact of the projectile from the target, the uptime of the device, etc. Distinguish between discrete and continuous random variables. Discrete A random variable is called, the possible values ​​of which form a countable set, finite or infinite (i.e., a set whose elements can be numbered).

continuous A random variable is called, the possible values ​​of which continuously fill some finite or infinite interval numerical axis. The number of values ​​of a continuous random variable is always infinite.

Random variables will be denoted capital letters the end of the Latin alphabet: X, Y, ...; values ​​of a random variable - in lowercase letters: X, y... . Thus, X Denotes the entire set of possible values ​​of a random variable, and X - Some specific meaning.

distribution law A discrete random variable is a correspondence given in any form between the possible values ​​of a random variable and their probabilities.

Let the possible values ​​of the random variable X Are . As a result of the test, the random variable will take one of these values, i.e. One event from a complete group of pairwise incompatible events will occur.

Let also the probabilities of these events be known:

Distribution law of a random variable X It can be written in the form of a table called Near distribution Discrete random variable:

The distribution series is equal (normalization condition).

Example 3.1. Find the distribution law of a discrete random variable X - the number of occurrences of the "eagle" in two coin tosses.

The distribution function is a universal form of setting the distribution law for both discrete and continuous random variables.

The distribution function of a random variableX The function is called F(X), Defined on the whole number line as follows:

F(X)= P(X< х ),

i.e. F(X) there is a probability that the random variable X Takes on a value less than X.

The distribution function can be represented graphically. For a discrete random variable, the graph has a stepped form. Let's build, for example, a graph of the distribution function of a random variable given by the following series (Fig. 3.1):

Rice. 3.1. Graph of the distribution function of a discrete random variable

Jumps of the function occur at points corresponding to the possible values ​​of the random variable, and are equal to the probabilities of these values. At break points, the function F(X) is continuous on the left.

The plot of the distribution function of a continuous random variable is a continuous curve.

X

Rice. 3.2. Graph of the distribution function of a continuous random variable

The distribution function has the following obvious properties:

1) , 2) , 3) ,

4) at .

We will call an event consisting in the fact that a random variable X Takes on a value X, Belonging to some semi-closed interval A£ X< B, By hitting a random variable on the interval [ A, B).

Theorem 3.1. The probability of a random variable falling into the interval [ A, B) is equal to the increment of the distribution function on this interval:

If we decrease the interval [ A, B), Assuming that , then in the limit, formula (3.1) instead of the probability of hitting the interval gives the probability of hitting the point, i.e., the probability that the random variable takes on the value A:

If the distribution function has a discontinuity at the point A, Then the limit (3.2) is equal to the jump value of the function F(X) at the point X=A, That is, the probabilities that the random variable will take on the value A (Fig. 3.3, A). If the random variable is continuous, i.e., the function is continuous F(X), then the limit (3.2) zero(Fig. 3.3, B)

Thus, the probability of any particular value of a continuous random variable is zero. However, this does not mean that the event is impossible. X=A, It only says that the relative frequency of this event will tend to zero with an unlimited increase in the number of tests.

A) B)

Rice. 3.3. Distribution function jump

For continuous random variables, along with the distribution function, another form of specifying the distribution law is used - the distribution density.

If is the probability of hitting the interval , then the ratio characterizes the density with which the probability is distributed in the vicinity of the point X. The limit of this relation at , i.e. e. derivative, is called Distribution density(density of probability distribution, probability density) of a random variable X. We agree to denote the distribution density

.

Thus, the distribution density characterizes the probability that a random variable will fall into the vicinity of the point X.

The plot of the distribution density is called crooked racesDefinitions(Figure 3.4).

Rice. 3.4. Distribution density type

Based on the definition and properties of the distribution function F(X), it is easy to establish the following properties of the distribution density F(X):

1) F(X)³0

2)

3)

4)

For a continuous random variable, due to the fact that the probability of hitting a point is zero, the following equalities hold:

Example 3.2. Random value X Specified by the distribution density

Required:

A) find the value of the coefficient A;

B) find the distribution function;

C) find the probability of a random variable falling into the interval (0, ).

The distribution function or distribution density completely describes a random variable. Often, however, when solving practical problems, there is no need for complete knowledge of the law of distribution, it is enough to know only some of it. character traits. To do this, in the theory of probability, numerical characteristics of a random variable are used, expressing various properties of the distribution law. The main numerical characteristics are MathematicalExpectation, variance and standard deviation.

Expected value Characterizes the position of a random variable on the number axis. This is some average value of a random variable around which all its possible values ​​are grouped.

Mathematical expectation of a random variable X Symbolized M(X) or T. The mathematical expectation of a discrete random variable is the sum of paired products of all possible values ​​of the random variable and the probabilities of these values:

The mathematical expectation of a continuous random variable is determined using an improper integral:

Based on the definitions, it is easy to verify the validity of the following properties of the mathematical expectation:

1. (mathematical expectation of a non-random variable WITH Equal to the most non-random value).

2. If ³0, then ³0.

4. If and independent, That .

Example 3.3. Find the mathematical expectation of a discrete random variable given by a series of distributions:

Solution.

=0×0.2 + 1×0.4 + 2×0.3 + 3×0.1=1.3.

Example 3.4. Find the mathematical expectation of a random variable given by the distribution density:

.

Solution.

Dispersion and standard deviation They are characteristics of the dispersion of a random variable, they characterize the spread of its possible values ​​relative to the mathematical expectation.

dispersion D(X) random variable X The mathematical expectation of the squared deviation of a random variable from its mathematical expectation is called. For a discrete random variable, the variance is expressed by the sum:

(3.3)

And for continuous - integral

(3.4)

The variance has the dimension of the square of a random variable. scattering characteristic, Matching in sizeStee with random variable, is the standard deviation.

Dispersion properties:

1) are constant. In particular,

3)

In particular,

Note that the calculation of the variance by formula (3.5) often turns out to be more convenient than by formula (3.3) or (3.4).

The value is called covariance random variables.

If , then the value

called Correlation coefficient random variables.

It can be shown that if , then the quantities are linearly dependent: where

Note that if they are independent, then

Example 3.5. Find the variance of a random variable given by the distribution series from Example 1.

Solution. To calculate the variance, you need to know the mathematical expectation. For a given random variable above, it was found: M=1.3. We calculate the variance using the formula (3.5):

Example 3.6. The random variable is given by the distribution density

Find the variance and standard deviation.

Solution. We first find the mathematical expectation:

(as an integral of an odd function over a symmetric interval).

Now we calculate the variance and standard deviation:

1. Binomial distribution. Random value , equal to the number"SUCCESS" in the Bernoulli scheme has a binomial distribution: , .

The mathematical expectation of a random variable distributed according to the binomial law is

.

The variance of this distribution is .

2. Poisson distribution ,

Mathematical expectation and variance of a random variable with Poisson distribution , .

The Poisson distribution is often used when we are dealing with the number of events that occur in a span of time or space, such as the number of cars arriving at a car wash in an hour, the number of machine stops per week, the number of traffic accidents, etc.

The random variable has Geometric distribution with parameter if takes values ​​with probabilities . A random variable with such a distribution makes sense Numbers of the first successful test in the Bernoulli scheme with the probability of success . The distribution table looks like:

3. Normal distribution. The normal law of probability distribution occupies a special place among other distribution laws. In probability theory, it is proved that the probability density of the sum of independent or Weakly dependent, uniformly small (i.e., playing approximately the same role) terms with an unlimited increase in their number approaches arbitrarily close to normal law distribution, regardless of what distribution laws these terms have (central limit theorem A. M. Lyapunova).

Distribution density probabilities X call the function f(x) is the first derivative of the distribution function F(x):

The concept of the probability distribution density of a random variable X for a discrete quantity is not applicable.

Probability density f(x) is called the differential distribution function:

Property 1. The distribution density is a non-negative value:

Property 2. Improper integral of the distribution density in the range from to equal to one:

Example 1.25. Given the distribution function of a continuous random variable X:

f(x).

Solution: The distribution density is equal to the first derivative of the distribution function:

1. Given the distribution function of a continuous random variable X:

Find the distribution density.

2. The distribution function of a continuous random variable is given X:

Find the distribution density f(x).

1.3. Numerical characteristics continuous random

quantities

Expected value continuous random variable X, the possible values ​​of which belong to the entire axis Oh, is determined by the equality:

It is assumed that the integral converges absolutely.

a,b), That:

f(x) is the distribution density of the random variable.

Dispersion continuous random variable X, the possible values ​​of which belong to the entire axis, is determined by the equality:

special case. If the values ​​of the random variable belong to the interval ( a,b), That:

The probability that X will take values ​​belonging to the interval ( a,b), is determined by the equality:

.

Example 1.26. Continuous random variable X

Find the mathematical expectation, variance and probability of hitting a random variable X in the interval (0; 0.7).

Solution: The random variable is distributed over the interval (0,1). Let us define the distribution density of a continuous random variable X:

a) Mathematical expectation :

b) Dispersion

V)

Tasks for independent work:

1. Random variable X given by the distribution function:

M(x);

b) dispersion D(x);

X into the interval (2,3).

2. Random variable X

Find: a) mathematical expectation M(x);

b) dispersion D(x);

c) determine the probability of hitting a random variable X in the interval (1; 1.5).

3. Random value X is given by the integral distribution function:

Find: a) mathematical expectation M(x);

b) dispersion D(x);

c) determine the probability of hitting a random variable X in the interval.

1.4. Laws of distribution of a continuous random variable

1.4.1. Uniform distribution

Continuous random variable X has a uniform distribution on the interval [ a,b], if on this segment the density of the probability distribution of a random variable is constant, and outside it is equal to zero, i.e.:

Rice. 4.

; ; .

Example 1.27. A bus of some route moves uniformly with an interval of 5 minutes. Find the probability that a uniformly distributed random variable X– the waiting time for the bus will be less than 3 minutes.

Solution: Random value X- uniformly distributed over the interval .

Probability Density: .

In order for the waiting time not to exceed 3 minutes, the passenger must arrive at the bus stop within 2 to 5 minutes after the departure of the previous bus, i.e. random value X must fall within the interval (2;5). That. desired probability:

Tasks for independent work:

1. a) find the mathematical expectation of a random variable X distributed uniformly in the interval (2; 8);

b) find the variance and standard deviation of a random variable X, distributed uniformly in the interval (2;8).

2. The minute hand of an electric clock jumps at the end of each minute. Find the probability that at a given moment the clock will show the time that differs from the true time by no more than 20 seconds.

1.4.2. The exponential (exponential) distribution

Continuous random variable X is exponentially distributed if its probability density has the form:

where is the parameter of the exponential distribution.

Thus

Rice. 5.

Numerical characteristics:

Example 1.28. Random value X- the operating time of the light bulb - has an exponential distribution. Determine the probability that the lamp will last at least 600 hours if the average lamp life is 400 hours.

Solution: According to the condition of the problem, the mathematical expectation of a random variable X equals 400 hours, so:

;

The desired probability , where

Finally:

Tasks for independent work:

1. Write the density and distribution function of the exponential law, if the parameter .

2. Random variable X

Find the mathematical expectation and variance of a quantity X.

3. Random value X given by the probability distribution function:

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

1.4.3. Normal distribution

normal is called the probability distribution of a continuous random variable X, whose density has the form:

Where A– mathematical expectation, – standard deviation X.

The probability that X will take a value belonging to the interval:

, Where

is the Laplace function.

A distribution that has ; , i.e. with a probability density called standard.

Rice. 6.

The probability that the absolute value deviated is less than positive number :

.

In particular, when a= 0 equality is true:

Example 1.29. Random value X distributed normally. Standard deviation . Find the probability that the deviation of a random variable from its mathematical expectation in absolute value will be less than 0.3.

Solution: .

Tasks for independent work:

1. Write the probability density of the normal distribution of a random variable X, knowing that M(x)= 3, D(x)= 16.

2. Mathematical expectation and standard deviation of a normally distributed random variable X are 20 and 5, respectively. Find the probability that as a result of the test X will take the value contained in the interval (15;20).

3. Random measurement errors are subject to the normal law with standard deviation mm and mathematical expectation a= 0. Find the probability that the error of at least one of 3 independent measurements does not exceed 4 mm in absolute value.

4. Some substance is weighed without systematic errors. Random weighing errors are subject to the normal law with a standard deviation r. Find the probability that the weighing will be carried out with an error not exceeding 10 g in absolute value.

mathematical expectation discrete random variable is called:

In the case of an infinite set of values, there is a series on the right side of (4.4), and we will consider only those values ​​of X for which this series converges absolutely.

M(X) is the average expected value of the random variable. It has the following properties:

1) M(C)=C, where C=const

2) M(CX)=CM(X) (4.5)

3) M (X+Y)=M(X)+M(Y), for any X and Y.

4) M (XY)=M (X)M(Y) if X and Y are independent.

To estimate the degree of dispersion of the values ​​of a random variable around its mean value M(X)= A concepts are introduced dispersionD(X) and the mean square (standard) deviation . dispersion is called the expectation of the square of the difference (X- ), those. :

D(X)=M(X- ) 2 = p i ,

Where =M(X); defined as Square root from dispersion, i.e. .

To calculate the variance, use the formula:

(4.6)

Properties of variance and standard deviation:

1) D(C)=0, where C=const

2) D(CX)=C 2 D(X), (CX)= çCç(X) (4.7)

3) D(X+Y) =D(X)+D(Y),

if X and Y are independent.

The dimension of the quantities and coincides with the dimension of the random variable X itself, and the dimension of D(X) is equal to the square of the dimension of the random variable X.

4.3. Mathematical operations on random variables.

Let the random variable X take values ​​with probabilities and the random variable Y take values ​​with probabilities values ​​of a random variable X. Therefore, its distribution law has the form table 4.2:

Table 4.2

			...
			...

Square random variable X, i.e. , is a new random variable that, with the same probabilities as the random variable X, takes values ​​equal to the squares of its values.

Sum random variables X and Y is a new random variable that takes all values ​​of the form with probabilities expressing the probability that the random variable X takes the value and Y - the value , that is

(4.8)

If the random variables X and Y are independent, then:

The difference and product of random variables X and Y are defined similarly.

Difference random variables X and Y is a new random variable that takes all values ​​of the form , and work- all values ​​of the form with probabilities determined by the formula (4.8), and if the random variables X and Y are independent, then by the formula (4.9).

4.4. Bernoulli and Poisson distributions.

Consider a sequence of n identical retests that satisfy the following conditions:

1. Each trial has two outcomes, called success and failure.

These two outcomes are mutually incompatible and opposite events.

2. The probability of success, denoted p, remains constant from trial to trial. The probability of failure is denoted by q.

3. All n trials are independent. This means that the probability of an event occurring in any of the n repeated trials does not depend on the results of other trials.

The probability that in n independent repeated trials, in each of which the probability of occurrence of an event is equal to , the event will occur exactly m times (in any sequence), is equal to

(4.10)

Expression (4.10) is called the Bernoulli formula.

The probabilities that the event will occur:

a) less than m times,

b) more than m times,

c) at least m times,

d) no more than m times - are found, respectively, according to the formulas:

Binomial is the law of distribution of a discrete random variable X - the number of occurrences of an event in n independent trials, in each of which the probability of the event occurring is equal to p; the probabilities of possible values ​​X = 0,1,2,..., m,...,n are calculated using the Bernoulli formula (Table 4.3).

Table 4.3

Number of successes X=m				...	m	...	n
Probability P				...		...

Since the right side of formula (4.10) represents the general term of the binomial expansion, this distribution law is called binomial. For a random variable X, distributed according to the binomial law, we have.

Expected value

Dispersion continuous random variable X, the possible values ​​of which belong to the entire axis Ox, is determined by the equality:

Service assignment. Online calculator designed to solve problems in which either distribution density f(x) , or distribution function F(x) (see example). Usually in such tasks it is required to find mathematical expectation, standard deviation, plot the functions f(x) and F(x).

Instruction. Select the type of input data: distribution density f(x) or distribution function F(x) .

The distribution density f(x) is given:

The distribution function F(x) is given:

A continuous random variable is defined by a probability density
(Rayleigh distribution law - used in radio engineering). Find M(x) , D(x) .

The random variable X is called continuous , if its distribution function F(X)=P(X< x) непрерывна и имеет производную.
The distribution function of a continuous random variable is used to calculate the probabilities of a random variable falling into a given interval:
P(α< X < β)=F(β) - F(α)
moreover, for a continuous random variable, it does not matter whether its boundaries are included in this interval or not:
P(α< X < β) = P(α ≤ X < β) = P(α ≤ X ≤ β)
Distribution density continuous random variable is called the function
f(x)=F'(x) , derivative of the distribution function.

Distribution Density Properties

1. The distribution density of a random variable is non-negative (f(x) ≥ 0) for all values ​​of x.
2. Normalization condition:

The geometric meaning of the normalization condition: the area under the distribution density curve is equal to one.
3. The probability of hitting a random variable X in the interval from α to β can be calculated by the formula

Geometrically, the probability that a continuous random variable X falls into the interval (α, β) is equal to the area of ​​the curvilinear trapezoid under the distribution density curve based on this interval.
4. The distribution function is expressed in terms of density as follows:

The distribution density value at the point x is not equal to the probability of taking this value; for a continuous random variable, we can only talk about the probability of falling into a given interval. Let )