- •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
Exercises for Homework 9
9.11. Compose the law of distribution of probabilities of the number of appearances of the event A in three independent trials if the probability of appearance of the event is 0,6 for each trial.
9.12. Let X be a random variable equal to the number of boys in families with five children. Assume that probabilities of births of both boy and girl are the same. Find the law of distribution of the random variable X. Find the probabilities of the following events:
(a) there are 2-3 boys in a family;
(b) no more than three boys;
(c) more than 1 boy.
The answer: a) 5/8; b) 13/16; c) 13/16.
9.13. A factory has sent 5000 suitable details to its warehouse. The probability that a detail is broken during a transportation is 0,0002. Find the probability that 3 non-suitable details will be arrived at the warehouse.
The answer: 0,06.
9.14. Suppose that the number of accidents occurring on a highway each day is a Poisson random variable with parameter = 3.
(a) Find the probability that 3 or more accidents occur today.
(b) Repeat part (a) under the assumption that at least 1 accident occurs today.
The answer: a) 0,577; b) 0,627.
9.15. A hunter shoots on a game before the first hit, but he managed to make no more than four shots. The probability of hit by him at one shot is 0,9.
(а) Find the law of distribution of a random variable X – the number of misses;
(b) Find the probability of the following events: X < 2, X 3, 1 < X 3 (hunter – охотник; game – дичь).
The answer: b) 0,99; 0,9999; 0,0099.
9.16. There are 11 students in a group, and 5 of them are girls. Compose the law of distribution of the random variable X – the number of girls from randomly selected three students.
9.17. There are 8 pencils in a box, and 5 of them are green. 3 pencils are randomly taken from the box.
(a) Find the law of distribution of a random variable X equal to the number of green pencils among taken.
(b) Find the probability of the event: 0 < X 2.
The answer: b) 45/56.
9.18. There are 20 products in a set, and 4 of them are defective. 3 products are randomly chosen for checking their quality. Compose the law of distribution of a random variable X – the number of defective products contained in the sample.
9.19. The probability of successful passing an exam by the first student is 0,7, and by the second – 0,8. Compose the law of distribution of a random variable X – the number of the students successfully passed the exam if each of them can retake only once the exam if he didn’t pass it at the first time. Find the mathematical expectation of the random variable X.
The answer: M(X) = 1,87.
9.20. A discrete random variable X is given by the following law of distribution:
xi |
1 |
X2 |
x3 |
3 |
pi |
0,1 |
P2 |
0,5 |
0,1 |
Determine x2, x3, p2, if it is known that M(X) = 4, M(X 2) = 20,2.
The answer: x2 = 2; x3 = 6 or x2 = 7; x3 = 3.
L E C T U R E 10