Familiarizing students with the basic concepts, theorems and methods of reasoning related to the probability theory.
Prerequisites
Getting a pass in Mathematical Analysis 1 and 2.
Scope
Lecture
1. Events and the probability
The revision of combinatorics. The classical definition of the probability. (2 hrs.)
The general definition of the probability. The definition and examples of the probability space and the event. Basic properties of the probability. (3 hrs.)
The geometrical probability. The conditional probability, the law of total probability and Bayes’ rule. (3 hrs.)
The independence of events. The Bernoulli scheme, the most likely number of successes in the Bernoulli scheme. (2 hrs.)
2. Random variables and their distributions
The definition, examples and properties of the random variable. The distribution of the random variable. The cumulative distribution function of the random variable and its properties. The cumulative distribution function and types of distributions. (4 hrs.)
Absolutely continuous and discrete distributions. The probability density function and its property. Overview of the most important absolutely continuous and discrete distributions. Mixed distributions. The independence of random variables. (4 hrs.)
Multidimensional random variables. The joint and marginal distributions, multidimensional and marginal cumulative distribution functions, marginal probability density functions. Connections with independent random variables. Distributions of sums of independent random variables. (3 hrs.)
3. The expectation and moments of random variables
The expectation and moments of a random variable. Examples of basic absolutely continuous and discrete distributions. The expectation and moments of random variables of mixed distribution, basic properties and interpretations. The variance and the standard deviation of random variables, basic properties and interpretation. (4 hrs.)
The concept of the covariance and the correlation coefficient of random variables, their connections with independent random variables. Parameters of random vectors. The multidimensional normal distribution. (2 hrs.)
4. Limit theorems
Chebyshev‘s inequality, the weak and strong law of large numbers, the central limit theorem and their applications. (3 hrs.)
Class
1. Events and the probability
The binomial coefficient and its interpretation. The use of basic combinatorial schemes to exercises related to the classical definition of the probability. (4 hrs.)
Determination of elementary events and events. Basic properties of the probability. (2 hrs.)
Exercises that use the geometric probability, the conditional probability, the law of total probability and Bayes’ rule. (2 hrs.)
Checking the independence of events. The calculation of probabilities of events with the assumption of independence. Exercises that use the Bernoulli scheme. (2 hrs.)
Colloquium (2 hrs.)
2. Random variables and their distributions, the expectation and moments of random variables
The verification whether some functions are random variables, cumulative distribution functions of some random variables. The determination of the cumulative distribution function of a random variable. The analysis of the distribution of a random variable on the basis of the cumulative distribution function. The verification whether some functions are probability density functions. The application of different types of discrete and absolutely continuous distributions in mathematical models. The application of normal distribution in exercises. (7 hrs.)
The determination of the joint and marginal distributions of two-dimensional random vectors using the tabular method. The determination of two-dimensional and marginal cumulative distribution functions, marginal probability density functions. The verification of the independence of random variables. Distributions of sums of independent random variables. (3 hrs.)
The determination of the expectation, moments and the variance of random variables. The properties of the expectation and the variance. The application in exercises. Calculations of the covariance and the correlation coefficient of random variables and their connections with the independence. The parameters of two-dimensional random vectors and two-dimensional normal distribution. (4 hrs.)
3. Limit theorems
The application of Chebyshev‘s inequality to estimate the probability of random variables. The application of the law of large numbers and the central limit theorem in exercises. (2 hrs.)
Colloquium (2 hrs.)
Teaching methods
A traditional lecture. Solving previously given tasks (exercises and short proofs) during the classes.
Learning outcomes and methods of theirs verification
Outcome description
Outcome symbols
Methods of verification
The class form
Assignment conditions
1. Checking the level of preparation of students and their activity during the classes.
2. Two colloquia with tasks of varying difficulty which allow to assess whether students have reached a minimum level of learning outcomes.
3. The exam in the form of a multiple-choice test, consisting of several dozen statements that require the verification on the basis of the acquired knowledge. The verification of statements is connected with the use of the theory or making simple calculations. The possible answers are Yes or No. The student may receive +1,-1 or 0 points for each statement.
To pass the class it is necessary to get passing scores in two colloquia. To take the exam it is necessary to pass the class. In order to pass the course it is necessary to get passing score in the exam. The final course grade is based on graded components: the class grade – 50% and the exam grade – 50%.
Recommended reading
1. J. K. Misiewicz, Wykłady z rachunku prawdopodobieństwa z zadaniami, SCRIPT, Warszawa 2005.
2. J. Jakubowski, R. Sztencel, Wstęp do teorii prawdopodobieństwa, SCRIPT, Warszawa 2000.
3. T. Inglot, T. Ledwina, Z. Ławniczak, Materiały do ćwiczeń z rachunku prawdopodobieństwa i statystyki matematycznej, PWR, Wrocław 1984.
4. A. E. Plucińscy, Elementy probabilistyki, PWN, Warszawa 1982.
5. G. Grimmett, D. Welsh, Probability: an introduction, Oxford University Press, 1986.
6. G. Roussas, Introduction to probability, Elsevier Science, 2006.
Further reading
1. J. Jakubowski, R. Sztencel, Rachunek prawdopodobieństwa dla (prawie) każdego, SCRIPT, Warszawa 2002.
2. W. Krysicki, J. Bartos, W. Dyczka, K. Królikowska, M. Wasilewski, Rachunek prawdopodobieństwa i statystyka matematyczna w zadaniach, część I, PWN, Warszawa 1999.
3. A. Plucińska, E. Pluciński, Zadania z probabilistyki, PWN, Warszawa 1983.
Notes
Modified by dr Robert Dylewski, prof. UZ (last modification: 20-09-2019 09:29)
