is a continuous random variable. Continuous random variable

§ 3. RANDOM VALUES

3. Continuous random variables.

In addition to discrete random variables, the possible values of which form a finite or infinite sequence of numbers that do not completely fill any interval, there are often random variables whose possible values form a certain interval. An example of such a random variable is the deviation from the nominal of a certain size of a part with a properly established technological process. This kind of random variables cannot be specified using the probability distribution law p(x). However, they can be specified using the probability distribution function F(x). This function is defined in exactly the same way as in the case of a discrete random variable:

Thus, here too the function F(x) defined on the whole number axis, and its value at the point X is equal to the probability that the random variable will take on a value less than X.
Formula () and properties 1° and 2° are valid for the distribution function of any random variable. The proof is carried out similarly to the case of a discrete quantity.
The random variable is called continuous, if for it there exists a non-negative piecewise-continuous function* that satisfies for any values x equality
Based on the geometric meaning of the integral as an area, we can say that the probability of fulfilling the inequalities is equal to the area of a curvilinear trapezoid with a base bounded above by a curve (Fig. 6).
Since , and based on the formula ()
, then
Note that for a continuous random variable, the distribution function F(x) continuous at any point X, where the function is continuous. This follows from the fact that F(x) is differentiable at these points.
Based on the formula (), assuming x 1 =x, , we have

Due to the continuity of the function F(x) we get that

Hence

Thus, the probability that a continuous random variable can take on any single value of x is zero.
It follows from this that the events consisting in the fulfillment of each of the inequalities
, , ,
They have the same probability, i.e.

Indeed, for example,

as

Comment. As we know, if an event is impossible, then the probability of its occurrence is zero. In the classical definition of probability, when the number of test outcomes is finite, the reverse proposition also takes place: if the probability of an event is zero, then the event is impossible, since in this case none of the test outcomes favors it. In the case of a continuous random variable, the number of its possible values is infinite. The probability that this value will take on any particular value x 1 as we have seen, is equal to zero. However, it does not follow from this that this event is impossible, since as a result of the test, the random variable can, in particular, take on the value x 1. Therefore, in the case of a continuous random variable, it makes sense to talk about the probability that the random variable falls into the interval, and not about the probability that it will take on a particular value.
So, for example, in the manufacture of a roller, we are not interested in the probability that its diameter will be equal to the nominal value. For us, the probability that the diameter of the roller does not go out of tolerance is important.

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. The improper integral of the distribution density in the range from to is equal to one:

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

f(x).

Decision: 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 of 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), then:

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), then:

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).

Decision: 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

in)

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 value 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.

Decision: 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.

Decision: 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 value 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 of the deviation is less than a 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.

Decision: .

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.

The distribution function in this case, according to (5.7), will take the form:

where: m is the mathematical expectation, s is the standard deviation.

The normal distribution is also called Gaussian after the German mathematician Gauss. The fact that a random variable has a normal distribution with parameters: m,, is denoted as follows: N (m, s), where: m =a =M ;

Quite often, in formulas, the mathematical expectation is denoted by a . If a random variable is distributed according to the law N(0,1), then it is called a normalized or standardized normal value. The distribution function for it has the form:

The graph of the density of the normal distribution, which is called the normal curve or Gaussian curve, is shown in Fig. 5.4.

Rice. 5.4. Normal distribution density

Determining the numerical characteristics of a random variable by its density is considered on an example.

Example 6.

A continuous random variable is given by the distribution density: .

Determine the type of distribution, find the mathematical expectation M(X) and the variance D(X).

Comparing the given distribution density with (5.16), we can conclude that the normal distribution law with m =4 is given. Therefore, mathematical expectation M(X)=4, variance D(X)=9.

Standard deviation s=3.

The Laplace function, which has the form:

is related to the normal distribution function (5.17), by the relation:

F 0 (x) \u003d F (x) + 0.5.

The Laplace function is odd.

Ф(-x)=-Ф(x).

The values of the Laplace function Ф(х) are tabulated and taken from the table according to the value of x (see Appendix 1).

The normal distribution of a continuous random variable plays an important role in the theory of probability and in the description of reality; it is very widespread in random natural phenomena. In practice, very often there are random variables that are formed precisely as a result of the summation of many random terms. In particular, the analysis of measurement errors shows that they are the sum of various kinds of errors. Practice shows that the probability distribution of measurement errors is close to the normal law.

Using the Laplace function, one can solve problems of calculating the probability of falling into a given interval and a given deviation of a normal random variable.

RANDOM VALUES

Example 2.1. Random value X given by the distribution function

Find the probability that as a result of the test X will take values between (2.5; 3.6).

Decision: X in the interval (2.5; 3.6) can be determined in two ways:

Example 2.2. At what values of the parameters BUT and AT function F(x) = A + Be - x can be a distribution function for non-negative values of a random variable X.

Decision: Since all possible values of the random variable X belong to the interval , then in order for the function to be a distribution function for X, the property should hold:

Answer: .

Example 2.3. The random variable X is given by the distribution function

Find the probability that, as a result of four independent trials, the value X exactly 3 times will take a value belonging to the interval (0.25; 0.75).

Decision: Probability of hitting a value X in the interval (0.25; 0.75) we find by the formula:

Example 2.4. The probability of the ball hitting the basket in one throw is 0.3. Draw up the law of distribution of the number of hits in three throws.

Decision: Random value X- the number of hits in the basket with three throws - can take on the values: 0, 1, 2, 3. The probabilities that X

Example 2.5. Two shooters make one shot at the target. The probability of hitting it by the first shooter is 0.5, the second - 0.4. Write down the law of distribution of the number of hits on the target.

Decision: Find the law of distribution of a discrete random variable X- the number of hits on the target. Let the event be a hit on the target by the first shooter, and - hit by the second shooter, and - respectively, their misses.

Let us compose the law of probability distribution of SV X:

Example 2.6. 3 elements are tested, working independently of each other. Durations of time (in hours) of failure-free operation of elements have distribution density functions: for the first: F 1 (t) =1-e- 0,1 t, for the second: F 2 (t) = 1-e- 0,2 t, for the third one: F 3 (t) =1-e- 0,3 t. Find the probability that in the time interval from 0 to 5 hours: only one element will fail; only two elements will fail; all three elements fail.

Decision: Let's use the definition of the generating function of probabilities:

The probability that in independent trials, in the first of which the probability of occurrence of an event BUT equals , in the second, etc., the event BUT appears exactly once, is equal to the coefficient at in the expansion of the generating function in powers of . Let's find the probabilities of failure and non-failure, respectively, of the first, second and third element in the time interval from 0 to 5 hours:

Let's create a generating function:

The coefficient at is equal to the probability that the event BUT will appear exactly three times, that is, the probability of failure of all three elements; the coefficient at is equal to the probability that exactly two elements will fail; coefficient at is equal to the probability that only one element will fail.

Example 2.7. Given a probability density f(x) random variable X:

Find the distribution function F(x).

Decision: We use the formula:

Thus, the distribution function has the form:

Example 2.8. The device consists of three independently operating elements. The probability of failure of each element in one experiment is 0.1. Compile the law of distribution of the number of failed elements in one experiment.

Decision: Random value X- the number of elements that failed in one experiment - can take the values: 0, 1, 2, 3. Probabilities that X takes these values, we find by the Bernoulli formula:

Thus, we obtain the following law of the probability distribution of a random variable X:

Example 2.9. There are 4 standard parts in a lot of 6 parts. 3 items were randomly selected. Draw up the law of distribution of the number of standard parts among the selected ones.

Decision: Random value X- the number of standard parts among the selected ones - can take the values: 1, 2, 3 and has a hypergeometric distribution. The probabilities that X

where -- the number of parts in the lot;

-- the number of standard parts in the lot;

– number of selected parts;

-- the number of standard parts among those selected.

Example 2.10. The random variable has a distribution density

where and are not known, but , a and . Find and .

Decision: In this case, the random variable X has a triangular distribution (Simpson distribution) on the interval [ a, b]. Numerical characteristics X:

Hence, . Solving this system, we get two pairs of values: . Since, according to the condition of the problem, we finally have: .

Answer: .

Example 2.11. On average, for 10% of contracts, the insurance company pays the sums insured in connection with the occurrence of an insured event. Calculate the mathematical expectation and variance of the number of such contracts among four randomly selected ones.

Decision: The mathematical expectation and variance can be found using the formulas:

Possible values of SV (number of contracts (out of four) with the occurrence of an insured event): 0, 1, 2, 3, 4.

We use the Bernoulli formula to calculate the probabilities of a different number of contracts (out of four) for which the sums insured were paid:

The distribution series of CV (the number of contracts with the occurrence of an insured event) has the form:


	0,6561	0,2916	0,0486	0,0036	0,0001

Answer: , .

Example 2.12. Of the five roses, two are white. Write a distribution law for a random variable expressing the number of white roses among two taken at the same time.

Decision: In a sample of two roses, there may either be no white rose, or there may be one or two white roses. Therefore, the random variable X can take values: 0, 1, 2. The probabilities that X takes these values, we find by the formula:

where -- number of roses;

-- number of white roses;

– the number of simultaneously taken roses;

-- the number of white roses among those taken.

Then the law of distribution of a random variable will be as follows:

Example 2.13. Among the 15 assembled units, 6 need additional lubrication. Draw up the law of distribution of the number of units in need of additional lubrication, among five randomly selected from the total number.

Decision: Random value X- the number of units that need additional lubrication among the five selected - can take the values: 0, 1, 2, 3, 4, 5 and has a hypergeometric distribution. The probabilities that X takes these values, we find by the formula:

where -- the number of assembled units;

-- number of units requiring additional lubrication;

– the number of selected aggregates;

-- the number of units that need additional lubrication among the selected ones.

Then the law of distribution of a random variable will be as follows:

Example 2.14. Of the 10 watches received for repair, 7 need a general cleaning of the mechanism. Watches are not sorted by type of repair. The master, wanting to find a watch that needs cleaning, examines them one by one and, having found such a watch, stops further viewing. Find the mathematical expectation and variance of the number of hours watched.

Decision: Random value X- the number of units that need additional lubrication among the five selected - can take the following values: 1, 2, 3, 4. The probabilities that X takes these values, we find by the formula:

Then the law of distribution of a random variable will be as follows:

Now let's calculate the numerical characteristics of the quantity :

Answer: , .

Example 2.15. The subscriber has forgotten the last digit of the phone number he needs, but remembers that it is odd. Find the mathematical expectation and variance of the number of dials he made before hitting the desired number, if he dials the last digit at random and does not dial the dialed digit in the future.

Decision: Random variable can take values: . Since the subscriber does not dial the dialed digit in the future, the probabilities of these values are equal.

Let's compose a distribution series of a random variable:


	0,2

Let's calculate the mathematical expectation and variance of the number of dialing attempts:

Answer: , .

Example 2.16. The probability of failure during the reliability tests for each device of the series is equal to p. Determine the mathematical expectation of the number of devices that failed, if tested N appliances.

Decision: Discrete random variable X is the number of failed devices in N independent tests, in each of which the probability of failure is equal to p, distributed according to the binomial law. 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:

Example 2.17. Discrete random variable X takes 3 possible values: with probability ; with probability and with probability . Find and knowing that M( X) = 8.

Decision: We use the definitions of mathematical expectation and the law of distribution of a discrete random variable:

We find: .

Example 2.18. The technical control department checks products for standardity. The probability that the item is standard is 0.9. Each batch contains 5 items. Find the mathematical expectation of a random variable X- the number of batches, each of which contains exactly 4 standard products, if 50 batches are subject to verification.

Decision: In this case, all experiments conducted are independent, and the probabilities that each batch contains exactly 4 standard products are the same, therefore, the mathematical expectation can be determined by the formula:

where is the number of parties;

The probability that a batch contains exactly 4 standard items.

We find the probability using the Bernoulli formula:

Answer: .

Example 2.19. Find the variance of a random variable X– number of occurrences of the event A in two independent trials, if the probabilities of the occurrence of an event in these trials are the same and it is known that M(X) = 0,9.

Decision: The problem can be solved in two ways.

1) Possible CB values X: 0, 1, 2. Using the Bernoulli formula, we determine the probabilities of these events:

, , .

Then the distribution law X looks like:

From the definition of mathematical expectation, we determine the probability:

Let's find the variance of SW X:

2) You can use the formula:

Answer: .

Example 2.20. 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; 25).

Decision: Probability of hitting a normal random variable X on the section from to is expressed in terms of the Laplace function:

Example 2.21. Given a function:

At what value of the parameter C this function is the distribution density of some continuous random variable X? Find the mathematical expectation and variance of a random variable X.

Decision: In order for a function to be the distribution density of some random variable , it must be non-negative, and it must satisfy the property:

Hence:

Calculate the mathematical expectation using the formula:

Calculate the variance using the formula:

T is p. It is necessary to find the mathematical expectation and variance of this random variable.

Decision: The distribution law of a discrete random variable X - the number of occurrences of an event in independent trials, in each of which the probability of the occurrence of an event is , is called binomial. The mathematical expectation of the binomial distribution is equal to the product of the number of trials and the probability of the occurrence of the event A in one trial:

Example 2.25. Three independent shots are fired at the target. The probability of hitting each shot is 0.25. Determine the standard deviation of the number of hits with three shots.

Decision: Since three independent trials are performed, and the probability of occurrence of the event A (hit) in each trial is the same, we will assume that the discrete random variable X - the number of hits on the target - is distributed according to the binomial law.

The variance of the binomial distribution is equal to the product of the number of trials and the probabilities of occurrence and non-occurrence of an event in one trial:

Example 2.26. The average number of clients visiting the insurance company in 10 minutes is three. Find the probability that at least one customer arrives in the next 5 minutes.

Average number of customers arriving in 5 minutes: . .

Example 2.29. The waiting time for an application in the processor queue obeys an exponential distribution law with an average value of 20 seconds. Find the probability that the next (arbitrary) request will wait for the processor for more than 35 seconds.

Decision: In this example, the expectation , and the failure rate is .

Then the desired probability is:

Example 2.30. A group of 15 students holds a meeting in a hall with 20 rows of 10 seats each. Each student takes a seat in the hall at random. What is the probability that no more than three people will be in the seventh place in the row?

Decision:

Example 2.31.

Then according to the classical definition of probability:

where -- the number of parts in the lot;

-- the number of non-standard parts in the lot;

– number of selected parts;

-- the number of non-standard parts among the selected ones.

Then the distribution law of the random variable will be as follows.

Continuous random variables have an infinite number of possible values. Therefore, it is impossible to introduce a distribution series for them.

Instead of the probability that the random variable X will take on a value equal to x, i.e. p(X = x), consider the probability that X will take on a value less than x, i.e. P(X< х).

We introduce a new characteristic of random variables - the distribution function and consider its properties.

The distribution function is the most universal characteristic of a random variable. It can be defined for both discrete and continuous random variables:

F(x) = p(X< x).

Distribution function properties.

The distribution function is a non-decreasing function of its argument, i.e. if:

At minus infinity, the distribution function is zero:

At plus infinity, the distribution function is equal to one:

The probability of a random variable falling into a given interval is determined by the formula:

The function f(x), which is equal to the derivative of the distribution function, is called the probability density of a random variable X or the distribution density:

Let's express the probability of hitting the section b to c in terms of f(x). It is equal to the sum of the probability elements in this section, i.e. integral:

From here, we can express the distribution function in terms of the probability density:

Probability density properties.

The probability density is a non-negative function (since the distribution function is a non-decreasing function):

Density probably

sti is a continuous function.

The integral in infinite limits of the probability density is equal to 1:

The probability density has the dimension of a random variable.

Mathematical expectation and dispersion of a continuous random variable

The meaning of the mathematical expectation and variance remains the same as in the case of discrete random variables. The form of formulas for finding them changes by replacing:

Then we obtain formulas for calculating the mathematical expectation and dispersion of a continuous random variable:

Example. The distribution function of a continuous random variable is given by:

Find the value of a, the probability density, the probability of hitting the site (0.25-0.5), the mathematical expectation and the variance.

Since the distribution function F(x) is continuous, then for x = 1 ax2 = 1, hence a = 1.

The probability density is found as a derivative of the distribution function:

The calculation of the probability of hitting a given area can be done in two ways: using the distribution function and using the probability density.

1st way. We use the formula for finding the probability through the distribution function:
2nd way. We use the formula for finding the probability through the probability density:

Finding the mathematical expectation:

Finding the variance:

Uniform distribution

Consider a continuous random variable X, the possible values of which lie in a certain interval and are equally probable.

The probability density of such a random variable will be:

where c is some constant.

The probability density graph will be displayed as follows:

We express the parameter c in terms of b and c. To do this, we use the fact that the integral of the probability density over the entire region must be equal to 1:

Distribution density of a uniformly distributed random variable

Find the distribution function:

Distribution function of a uniformly distributed random variable

Let's plot the distribution function:

Let us calculate the mathematical expectation and variance of a random variable obeying a uniform distribution.

Then the standard deviation will look like:

Normal (Gaussian) distribution

A continuous random variable X is called normally distributed with parameters a, y > 0 if it has a probability density:

The distribution curve of a random variable has the form:

Test 2

Task 1. Compose the law of distribution of a discrete random variable X, calculate the mathematical expectation, variance and standard deviation of a random variable.

Option 1

QCD checks products for standardization. The probability that the item is standard is 0.7. 20 items tested. Find the law of distribution of the random variable X - the number of standard products among the tested ones. Calculate the mathematical expectation, variance and standard deviation of a random variable.

Option 2

There are 4 balls in the urn, on which points 2 are indicated; 4; 5; 5. A ball is drawn at random. Find the law of distribution of a random variable X - the number of points on it. Calculate the mathematical expectation, variance and standard deviation of a random variable.

Option 3

The hunter shoots the game until it hits, but can fire no more than three shots. The probability of hitting each shot is 0.6. Compose the law of distribution of the random variable X - the number of shots fired by the shooter. Calculate the mathematical expectation, variance and standard deviation of a random variable.

Option 4

The probability of exceeding the specified accuracy in the measurement is 0.4. Compose the law of distribution of a random variable X - the number of errors in 10 measurements. Calculate the mathematical expectation, variance and standard deviation of a random variable.

Option 5

The probability of hitting the target with one shot is 0.45. 20 shots fired. Compose the law of distribution of a random variable X - the number of hits. Calculate the mathematical expectation, variance and standard deviation of a random variable.

Option 6

Products of a certain factory contain 5% of the marriage. Make a distribution law for a random variable X - the number of defective products among five taken for good luck. Calculate the mathematical expectation, variance and standard deviation of a random variable.

Option 7

The parts needed by the assembler are in three of the five boxes. The assembler opens the boxes until he finds the right parts. Compose the law of distribution of a random variable X - the number of opened boxes. Calculate the mathematical expectation, variance and standard deviation of a random variable.

Option 8

An urn contains 3 black and 2 white balls. Sequential extraction of balls without return is carried out until black appears. Compose the law of distribution of a random variable X - the number of extracted balls. Calculate the mathematical expectation, variance and standard deviation of a random variable.

Option 9

The student knows 15 questions out of 20. There are 3 questions in the ticket. Compose the law of distribution of a random variable X - the number of questions known to the student in the ticket. Calculate the mathematical expectation, variance and standard deviation of a random variable.

Option 10

There are 3 light bulbs, each of which has a defect with a probability of 0.4. When turned on, the defective light bulb burns out and is replaced by another. Make a distribution law for a random variable X - the number of lamps tested. Calculate the mathematical expectation, variance and standard deviation of a random variable.

Task 2. The random variable X is given by the distribution function F(X). Find the distribution density, mathematical expectation, variance, and also the probability of a random variable falling into the interval (b, c). Construct graphs of the functions F(X) and f(X).

Option 1