Reshebnik Ryabushko. Solved IDZ 18.2, Option 1

Description

The following tasks are given:

1. Find the distribution law of the discrete SV X and its distribution function F(x). Calculate the mathematical expectation M(X), variance D(X) and standard deviation σ(X). Plot the distribution function F(x).

   1.1. A car must drive along a street that has four independently operating traffic lights. Each traffic light gives red and green signals at intervals of 2 minutes; CB X is the number of car stops on this street.

2. The distribution function F(x) of the SV X is given. Find the probability distribution density f(x), the mathematical expectation M(X), the dispersion D(X) and the probability of the SV X falling on the segment [a; b]. Draw graphs of the functions F(x) and f(x).

3. Solve the following problems:

   3.1. A roller made by an automatic machine is considered standard if the deviation of its diameter from the design size does not exceed 2 mm. Random deviations of roller diameters obey the normal law with a standard deviation of 1.6 mm and a mathematical expectation equal to 0. How many standard rollers (in percent) does the machine produce?

4. Solve the following problems:

   4.1. To determine the quality of the products produced by the plant, 2,500 products were selected at random. Among them there were 50 with defects. The frequency of manufacturing defective products is taken as an approximate value of the probability of manufacturing a defective product. Determine with what probability it is possible to guarantee that the absolute error allowed will not exceed 0.02.

Answer

1. To solve the problem, it is necessary to determine the distribution law and the distribution function of the discrete SV X. Knowing the distribution function, you can find the probability distribution density, and then the mathematical expectation, dispersion and standard deviation. Plot the distribution function F(x).

   For CB X we will use the following notation: X is the number of car stops on the street. Each traffic light emits red and green signals at intervals of 2 minutes, so CB X takes values ​​from 0 to 4 inclusive. The values ​​of SV X and their probabilities are given in the table:

   X	0	1	2	3	4
   P(X)	0,0625	0,25	0,375	0,25	0,0625

   Here P(X) is the probability that SV X will take the corresponding value.

   The distribution function F(x) for a discrete SV X is defined as the sum of the probabilities of values ​​of SV X less than or equal to x:

   F(x) = P(X ≤ x)

   Let's plot the distribution function F(x):

   Let's find the probability distribution density f(x). To do this we use the formula:

   f(x) = P(X = x)

   Table of f(x) values ​​for SV X:

   X	0	1	2	3	4
   f(x)	0,0625	0,1875	0,125	0,1875	0,0625

   The mathematical expectation M(X) and the variance D(X) for a discrete SV X are calculated using the formulas:

   M(X) = S_i=1ⁿ x_i P(X = x_i)

   D(X) = S_i=1ⁿ (x_i - M(X))² P(X = x_i)

   Calculating using these formulas, we get:

   M(X) = 1,5

   D(X) = 1,25

   The standard deviation σ(X) is equal to the root of the variance:

   σ(X) = √D(X) ≈ 1.12

2. The distribution function F(x) of the SV X is given. It is necessary to find the probability distribution density f(x), the mathematical expectation M(X), the dispersion D(X) and the probability of the SV X falling on the segment [a; b]. Draw graphs of the functions F(x) and f(x).

   Let the distribution function F(x) CB X be given by the following formula:

   F(x) = 1 - e^-λx, x ≥ 0

   where λ > 0 is the distribution parameter.

   The probability distribution density f(x) for SV X is calculated as the derivative of the distribution function:

   f(x) = F'(x) = λe^-λx, x ≥ 0

   Expectation M(X

   "Reshebnik Ryabushko. Solved IDZ 18.2, Option 1" is a digital product that is a collection of solved problems in mathematics from Individual Homework (IDZ) 18.2, Option 1. The solver contains detailed and understandable solutions to all problems, as well as a step-by-step explanation of each solutions.

   The product is designed in a beautiful and convenient HTML format, which makes it easy and quick to find the desired task and view its solution. The product is intended for schoolchildren and students who study mathematics and want to improve their knowledge and skills in this area.

   In addition, this workbook can be used as an example and help in solving similar problems, which can be useful in preparing for exams and tests in mathematics. Having detailed solutions allows you to understand the material faster and more effectively and increase your level of knowledge.

   Purchasing this digital product allows you to access solutions to problems at any convenient time and place, without having to carry heavy textbooks and problem books with you. The beautiful and convenient design of the product makes its use more pleasant and effective.

"Reshebnik Ryabushko. Solved IDZ 18.2, Option 1" is an electronic collection of solutions to problems in mathematics from Individual Homework (IDZ) 18.2, Option 1, which is intended for schoolchildren and students. The workbook provides detailed and clear solutions to all problems, including a step-by-step explanation of each solution.

This IDZ examines problems in probability theory and mathematical statistics. In particular, the solution book contains solutions to problems of finding the distribution law of a discrete random variable, its distribution function, probability distribution density, mathematical expectation, dispersion and standard deviation. Problems of finding the probability of a random variable falling on a given segment and assessing the quality of manufactured products are also considered.

The product is designed in a convenient HTML format, which makes it easy and quick to find the desired task and view its solution. Ryabushko's solution book is a useful tool for teaching mathematics and allows you to improve your understanding of the material and improve your performance at school or university.

***

Ryabushko's solver is a collection of solved problems in mathematics, probability and statistics. In this case, Individual Homework (IH) 18.2, Option 1, which includes several tasks, has been solved.

The first task is to find the distribution law and probability distribution function of a discrete random variable X, which is the number of car stops on a street with four traffic lights. It is also required to calculate the mathematical expectation, variance and standard deviation of the random variable X, as well as plot the distribution function F(x).

The second task is to find the probability distribution density f(x), the mathematical expectation and dispersion of the random variable X, as well as the probability of SV X falling on the segment [a; b]. It is also required to plot the distribution function F(x) and the distribution density f(x).

The third task is related to determining the number of standard rollers produced by the machine at a given standard deviation and the mathematical expectation of the deviation of the roller diameter from the design size.

The fourth task is to determine the probability that when sampling 2500 products from the plant's production, the proportion of defective products will not exceed a given absolute error.

***

1. Ryabushko's Solver is an excellent assistant for schoolchildren who want to solve problems quickly and easily.
2. The solved IDZ 18.2, Option 1 helped my child better understand the material and prepare for the exams.
3. With the help of Reshebnik Ryabushko, my child began to answer questions more confidently in class.
4. An excellent purchase for schoolchildren who want to improve their knowledge and skills in various subjects.
5. Ryabushko’s solution book is a reliable and accurate source of information that helps solve problems correctly.
6. A large selection of solved tasks in the Ryabushko Reshebnik allows schoolchildren to prepare for any tests and exams.
7. Ryabushko's solution book is a great way to increase your academic performance and confidence in your knowledge.

Peculiarities:

Reshebnik Ryabushko is an excellent assistant for schoolchildren who want to improve their knowledge and skills.

Solved IDZ 18.2, Option 1 is a great example of how the Ryabushko Solver helps students with their homework.

With Ryabushko's Reshebnik, my children began to better understand the material and get high marks in the lessons.

A very convenient format of Ryabushko's Reshebnik - all solutions are presented in a clear and easily accessible form.

With the help of Ryabushko's Reshebnik, my child began to feel more confident in the classroom and successfully cope with complex tasks.

Ryabushko's Resource Book is not just a list of answers, but a useful tool for learning and raising the level of knowledge.

Many thanks to the creators of Ryabushko's Reshebnik for such a useful and convenient product!