BSc: Probability And Statistics.f22

Probability and Statistics

• Course name: Probability and Statistics
• Code discipline: CSE206
• Subject area: Math

Short Description

The course is designed to provide Software Engineers and Computer Scientists by correct knowledge of basic (core) concepts, definitions, theoretical results and applied methods & techniques of Probability Theory and Mathematical Statistics. The main idea of the course is to study mathematical basis of modelling random experiments. The course includes constructing a probability space, a model of a random experiment, and its applications to practice. After that, random variables and their properties are considered. As examples of applying this theoretical background, limit theorems of probability theory are proved (law of large numbers, central limit theorem) and some elements of mathematical statistics are studied.

Course Topics

Course Sections and Topics

Section	Topics within the section
Basics of Probability	Sampling Procedures, Collection of Data Measures of Location: The Sample Mean and Median, Measures of Variability, Discrete and Continuous Data Sample Space, Events, Probability of an Event, Additive Rules, Conditional Probability, Bayes’ Rule Discrete Probability Distributions. Continuous Probability Distributions, Joint Probability Distributions Mean of a Random Variable, Variance and Covariance of Random Variables, Chebyshev’s Theorem
Some Probability Distributions	Binomial and Multinomial Distributions, Hypergeometric Distribution, Poisson Distribution and the Poisson Process Some Continuous Probability Distributions Uniform, Normal, Gamma, Exponential, Chi-Squared, Beta, Lognormal, Weibull Distributions
Basics of Statistics	Sampling Distribution of Means and the Central Limit Theorem. t-Distribution, F-Distribution,Quantile and Probability Plots Estimating the Mean, Proportion, Variance, Differences, Maximum Likelihood Testing a Statistical Hypothesis, Test for Independence, Test for Homogeneity Least Squares and the Fitted Model, Choice of a Regression Model, Analysis-of-Variance Approach

Intended Learning Outcomes (ILOs)

What is the main purpose of this course?

This calculus course will provide an opportunity for participants to:

• understand key principles involved in differentiation and integration of functions
• solve problems that connect small-scale (differential) quantities to large-scale (integrated) quantities
• become familiar with the fundamental theorems of Calculus
• get hands-on experience with the integral and derivative applications and of the inverse relationship between integration and differentiation.

ILOs defined at three levels

We specify the intended learning outcomes at three levels: conceptual knowledge, practical skills, and comprehensive skills.

Level 1: What concepts should a student know/remember/explain?

By the end of the course, the students should be able to ...

• know the probability function and its properties
• know the law of total probability and Bayes’ theorem
• explain the independence of events and of random variables
• know the different continuous distributions
• know the multivariate distributions for discrete and continuous cases
• know the maximum likelihood estimator method

Level 2: What basic practical skills should a student be able to perform?

By the end of the course, the students should be able to ...

• construct a mathematical model of a random experiment (probability space)
• calculate conditional probabilities
• use probability generating functions for discrete random variables
• find confidence intervals for parameters of a normal distribution
• estimate unknown parameters of distributions

Level 3: What complex comprehensive skills should a student be able to apply in real-life scenarios?

By the end of the course, the students should be able to ...

• find expected value, variance and other characteristics of a random variable
• apply limit theorems (law of large numbers and central limit theorem)
• find parameters of a simple linear regression

Grading

Course grading range

Grade	Range	Description of performance
A. Excellent	85-100	-
B. Good	70-84	-
C. Satisfactory	50-69	-
D. Fail	0-49	-

Course activities and grading breakdown

Activity Type	Percentage of the overall course grade
Midterm	20
Quizzes	28 (2 for each)
Final exam	50
In-class participation	7 (including 5 extras)

Recommendations for students on how to succeed in the course

• Participation is important. Attending lectures is the key to success in this course.
• Review lecture materials before classes to do well.
• Reading the recommended literature is obligatory, and will give you a deeper understanding of the material.

Resources, literature and reference materials

Open access resources

• Probability & statistics for engineers & scientists/Ronald E. Walpole ... [et al.] — 9th ed. p. cm. ISBN 978-0-321-62911-1

book

• Durrett Rick. (2019) Probability. Theory and Examples,

Closed access resources

• Suhov Y, Kelbert M (2005) Probability and Statistics by Example, Cambridge University Press

Software and tools used within the course

• No.

Activities and Teaching Methods

Teaching and Learning Methods within each section

Teaching Techniques	Section 1	Section 2	Section 3
Problem-based learning (students learn by solving open-ended problems without a strictly-defined solution)	1	1	1
Project-based learning (students work on a project)	0	0	0
Modular learning (facilitated self-study)	0	0	0
Differentiated learning (provide tasks and activities at several levels of difficulty to fit students needs and level)	1	1	1
Contextual learning (activities and tasks are connected to the real world to make it easier for students to relate to them)	0	0	0
Business game (learn by playing a game that incorporates the principles of the material covered within the course)	0	0	0
Inquiry-based learning	0	0	0
Just-in-time teaching	0	0	0
Process oriented guided inquiry learning (POGIL)	0	0	0
Studio-based learning	0	0	0
Universal design for learning	0	0	0
Task-based learning	0	0	0

Activities within each section

Learning Activities	Section 1	Section 2	Section 3
Lectures	1	1	1
Interactive Lectures	1	1	1
Lab exercises	1	1	1
Experiments	0	0	0
Modeling	0	0	0
Cases studies	0	0	0
Development of individual parts of software product code	0	0	0
Individual Projects	0	0	0
Group projects	0	0	0
Flipped classroom	0	0	0
Quizzes (written or computer based)	1	1	1
Peer Review	0	0	0
Discussions	1	1	1
Presentations by students	0	0	0
Written reports	0	0	0
Simulations and role-plays	0	0	0
Essays	0	0	0
Oral Reports	0	0	0

Formative Assessment and Course Activities

Ongoing performance assessment

Section 1

1. Each game of a match between two equal players can end with a victory of one of them with probability ${\textstyle 0{.}5}$ independently of the other games. Each victory yields one point, and the match is played until one of the players scores 6 points. Due to technical reasons the match was interrupted when the score was ${\textstyle 5:3}$ in favour of the first player. What do you think is a fair way to distribute the prize between the players?
2. Seventy numbers are chosen at random from integers ${\textstyle 1,2,3,\cdot ,100}$. What is the probability that the largest number chosen is ${\textstyle 98}$?
3. A hospital specialises in curing three types of diseases: ${\textstyle A}$, ${\textstyle B}$ and ${\textstyle C}$. On average, there are ${\textstyle 50\%}$ of patients who suffer from disease ${\textstyle A}$, ${\textstyle 30\%}$ of patients with disease ${\textstyle B}$, and ${\textstyle 20\%}$ of patients with disease ${\textstyle C}$ (each of the patients has exactly one of these diseases). The probabilities to fully recover from the diseases are equal to ${\textstyle 0{.}95}$, ${\textstyle 0{.}9}$ and ${\textstyle 0{.}85}$ respectively. A patient who came to the hospital recovered completely. What is the probability that he had disease ${\textstyle B}$?
4. A white ball is added into an urn that initially contained ${\textstyle n}$ balls. It is known that the probabilities of having ${\textstyle 0,1,2,\ldots n}$ white balls (at the start) in the urn are equal to each other. (a) One ball is taken at random from the urn. What is the probability that the ball is white? (b) The ball taken from the urn has turned out to be white. Find the most probable number of white balls that were in the urn from the start.

Section 2

1. Find the following integrals:
   • ${\textstyle \int {\frac {{\sqrt {4+x^{2}}}+2{\sqrt {4-x^{2}}}}{\sqrt {16-x^{4}}}}\,dx}$;
   • ${\textstyle \int 2^{2x}e^{x}\,dx}$;
   • ${\textstyle \int {\frac {dx}{3x^{2}-x^{4}}}}$.
2. Find the indefinite integral ${\textstyle \displaystyle \int x\ln \left(x+{\sqrt {x^{2}-1}}\right)\,dx}$.
3. Find the length of a curve given by ${\textstyle y=\ln \sin x}$, ${\textstyle {\frac {\pi }{4}}\leqslant x\leqslant {\frac {\pi }{2}}}$.

Section 3

1. Find limits of the following sequences or prove that they do not exist:
   • $a_{n}=n-{\sqrt {n^{2}-70n+1400}}$;
   • ${\textstyle d_{n}=\left({\frac {2n-4}{2n+1}}\right)^{n}}$;
   • ${\textstyle x_{n}={\frac {\left(2n^{2}+1\right)^{6}(n-1)^{2}}{\left(n^{7}+1000n^{6}-3\right)^{2}}}}$.

Final assessment

Section 1

1. Apply the appropriate differentiation technique to a given problem.
2. Find a derivative of a function
3. Apply Leibniz formula
4. Draw graphs of functions
5. Find asymptotes of a parametric function

Section 2

1. Apply the appropriate integration technique to the given problem
2. Find the value of the devinite integral
3. Calculate the area of the domain or the length of the curve

Section 3

1. Find a limit of a sequence
2. Find a limit of a function

The retake exam

Retakes will be run as a comprehensive exam, where the student will be assessed the acquired knowledge coming from the textbooks, the lectures, the labs, and the additional required reading material, as supplied by the instructor. During such comprehensive oral/written the student could be asked to solve exercises and to explain theoretical and practical aspects of the course.