- •Introduction
- •Basic concepts of probability theory
- •Classical definition of probability
- •Relative frequency
- •Geometric probabilities
- •Glossary
- •Exercises for Seminar 1
- •Exercises for Homework 1
- •Basic formulas of combinatorial analysis
- •Operations over events
- •Glossary
- •Exercises for Seminar 2
- •Exercises for Homework 2
- •Theorem of addition of probabilities of incompatible events
- •Complete group of events
- •Opposite events
- •Conditional probability
- •Theorem of multiplication of probabilities
- •Glossary
- •Exercises for Seminar 3
- •Exercises for Homework 3
- •Independent events
- •Where a is the appearance of at least one of the events a1, a2, …, An; .
- •Glossary
- •Exercises for Seminar 4
- •Exercises for Homework 4
- •Theorem of addition of probabilities of compatible events
- •Formula of total probability
- •Probability of hypotheses. Bayes’s formulas.
- •Glossary
- •Exercises for Seminar 5
- •Exercises for Homework 5
- •Repetition (recurrence) of trials. The Bernoulli formula
- •Local theorem of Laplace
- •Integral theorem of Laplace
- •Glossary
- •Exercises for Seminar 6
- •Exercises for Homework 6
- •Random variables. The law of distribution of a discrete random variable
- •A random variable is understood as a variable which as result of a trial takes one of the possible set of its values (which namely – it is not beforehand known).
- •Mathematical operations over random variables
- •(Mathematical) expectation of a discrete random variable
- •Dispersion of a discrete random variable
- •Glossary
- •Exercises for Seminar 7
- •Exercises for Homework 7
- •Distribution function of a random variable
- •Properties of a distribution function
- •Continuous random variables. Probability density
- •Properties of probability density
- •Glossary
- •Exercises for Seminar 8
- •Exercises for Homework 8
- •Basic laws of distribution of discrete random variables
- •1. Binomial law of distribution
- •2. The law of distribution of Poisson
- •3. Geometric distribution
- •4. Hypergeometric distribution
- •Glossary
- •Exercises for Seminar 9
- •Exercises for Homework 9
- •Basic laws of distribution of continuous random variables
- •1. The uniform law of distribution
- •2. Exponential law of distribution
- •3. Normal law of distribution
- •Glossary
- •Exercises for Seminar 10
- •Exercises for Homework 10
- •The law of large numbers and limit theorems
- •The central limit theorem
- •Glossary
- •Exercises for Seminar 11
- •Exercises for Homework 11
- •Mathematical statistics. Variation series and their characteristics
- •Numerical characteristics of variation series
- •Glossary
- •Exercises for Seminar 12
- •Exercises for Homework 12
- •Bases of the mathematical theory of sampling
- •Glossary
- •Exercises for Seminar 13
- •Exercises for Homework 13
- •Methods of finding of estimations
- •Notion of interval estimation
- •Glossary
- •Exercises for Seminar 14
- •Exercises for Homework 14
- •Testing of statistical hypotheses
- •Glossary
- •Exercises for Seminar 15
- •Exercises for Homework 15
- •Individual homeworks
- •Variant 1
- •Variant 2
- •Variant 3
- •Variant 4
- •Variant 5
- •Variant 6
- •Variant 7
- •Variant 8
- •Variant 9
- •Variant 10
- •Variant 11
- •Variant 12
- •Variant 13
- •Variant 14
- •Variant 15
- •Variant 16
- •Variant 17
- •Variant 18
- •Variant 19
- •Variant 20
- •Variant 21
- •Variant 22
- •Variant 23
- •Variant 24
- •Variant 25
- •Final exam trial tests (for self-checking)
- •Appendix
- •Values the functions and
- •List of the used books
- •Contents
Glossary
shell – артиллерийский снаряд; aim, sight – прицел
exhaustive – исчерпывающий; assignment – задание
ascending order – возрастающий порядок
mathematical expectation, mean value – математическое ожидание
rod – стержень; on the average – в среднем; deviation – отклонение
concentration – кучность; to strike – поражать
dispersion, variance – дисперсия, рассеяние
mean square deviation – среднее квадратическое отклонение
numerical characteristic – числовая характеристика
Exercises for Seminar 7
7.1. Two balls are chosen randomly from an urn containing 8 white, 4 black and 2 orange balls. Suppose that we win $2 for each black ball selected and we lose $1 for each white ball selected. Let X denote our winnings. What are the possible values of X, and what are the probabilities associated with each value?
7.2. The probability of working each of four combines without breakages during a certain time is equal to 0,9. Compose the law of distribution of a random variable X – the number of combines working trouble-free. Find the mathematical expectation, the dispersion and the mean square deviation of the random variable X.
The answer: M(X) = 3,6; D(X) = 0,36; (X) = 0,6.
7.3. The probability of birth of a boy in a family is equal to 0,515. Compose the law of distribution of a random variable X – the number of boys in families having four children. Find the mathematical expectation, the dispersion and the mean square deviation.
The answer: M(X) = 2,06; D(X) = 0,999; (X) = 1,0.
7.4. There are 6 masters of sports in a group of 10 sportsmen. One selects (under the circuit without replacement) 3 sportsmen. Compose the law of distribution of a random variable X – the number of masters of sports of the selected sportsmen. Find the mathematical expectation of the random variable X.
The answer: M(X) = 1,8.
7.5. A shooter makes shots in a target before the first hit. The probability of hit in the target at each shot is equal to 0,7. Compose the law of distribution of a random variable X – the number of shots made by the shooter. Find the most probable number of cartridges (patrons) given to the shooter.
The answer: k0 = 1.
7.6. The mathematical expectation of a random variable X is equal to 8. Find the mathematical expectation of the following random variables: a) X – 4; b) 3X + 4.
7.7. The dispersion of a random variable X is equal to 8. Find the dispersion of the following random variables: a) X – 2; b) 3X + 2.
7.8. Independent random variables X and Y have the following distributions:
-
X
2
4
6
Y
3
4
p
0,3
0,5
0,2
p
0,4
0,6
Compose the law of distribution of the random variable Z = X + Y. Find the mathematical expectation, the dispersion and the mean square deviation of the random variable Z.
The answer: M(Z) = 7,4; D(Z) = 2,2.
7.9. Find the mathematical expectation and the dispersion of random variable Z = 4X – 2Y if M(X) = 5, M(Y) = 3, D(X) = 4, D(Y) = 6. The random variables X and Y are independent.
The answer: M(Z) = 14; D(Z) = 88.
7.10. A total of 4 buses carrying 148 students from the same school arrives at a football stadium. The buses carry, respectively, 40, 33, 25, and 50 students. One of the students is randomly selected. Let X denote the number of students that were on the bus carrying this randomly selected student. One of the 4 bus drivers is also randomly selected. Let Y denote the number of students on his bus. Which of M(X) or M(Y) do you think is larger? Why? Compute M(X) and M(Y).