BSc: Probability And Statistics.f23

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

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	25
Tests	30 (2 x 15pts)
Final exam	40
Sudden quizes	12 (3 x 4pts, including 7 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 book

Closed access resources

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

Software and tools used within the course

• No.

Teaching Methodology: Methods, techniques, & activities

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. On average ${\textstyle 25\%}$ students subscribe to the newsletter. Determine the most probable number of subscribers out of (a) 100 students; (b) 103 students.
2. Two players are playing a match (that consists of several games), each of the games can finish in favour of the younger player with probability ${\textstyle 0{.}6}$ and in favour of the older player with probability ${\textstyle 0{.}4}$ The younger player has won exactly five games in the first eight games. What is the probability that he started the match with a defeat?
3. Find the range of variance for random variable ${\textstyle \eta }$ if its cumulative distribution function is given by ${\textstyle F_{\eta }(x)={\begin{cases}0,&x\leq 0,\\0{.}3,&0<x\leq 2,\\b,&2<x\leq 6,\\1,&x>6,\end{cases}}}$ if ${\textstyle b}$ is a parameter that belongs to ${\textstyle (0{.}3;1)}$.
4. Six people entered the lift at the ground floor of a nine-storied house. Find the expected value for (a) the number of stops where exactly one person gets off the lift; (b) the number of stops where exactly two persons leave the lift.
5. Find the expected value and variance of ${\textstyle a^{\xi }}$ given that ${\textstyle \xi \sim Bin(n,p)}$.
6. Random variable ${\textstyle Y}$ has a uniform distribution on interval ${\textstyle (a;b)}$, and ${\textstyle EY=VarY=3}$. Find ${\textstyle a}$ and ${\textstyle b}$.

Section 3

1. How many times does one have to flip a coin to get the results “heads”, “heads” in succession? Is the result going to change if we replace the sequence with “tails”, “heads”?
2. Forty three equally strong sportsmen take part in a ski race; 18 of them belong to club ${\textstyle A}$, 10 to club ${\textstyle B}$ and 15 to club ${\textstyle C}$. What is the average place for (a) the best participant from club ${\textstyle B}$; (b) the worst participant from club ${\textstyle B}$?
3. How much rolls does one need on average to get a sequence “6”, “6” when rolling a symmetric six-sided die? And if we change this sequence to “6”, “6”, “6”?
4. ${\textstyle \zeta }$ is the quantity of threes and ${\textstyle \eta }$ is the quantity of odd digits obtained when rolling a fair die ${\textstyle K}$ times. Find correlation coefficient between ${\textstyle \eta }$ and ${\textstyle \zeta }$.

Final assessment

Section 1

1. The probabilities for three students to pass the exam are equal to ${\textstyle {\frac {11}{12}}}$, ${\textstyle {\frac {11}{14}}}$ and ${\textstyle {\frac {18}{25}}}$ respectively. Determine the probability that at least one student passes the exam given that they pass or fail independently of each other.
2. One of 10-digit numbers in which digits go in non-increasing order is chosen at random. Find the probability that exactly 4 different digits are used in this number.
3. Two persons play a game. They take turns in rolling a 10-sided fair die. The first one wins as soon as he rolls 9 or 10, whereas the second one wins as soon as he gets no more than 4. (The game goes on until one of the player’s winning conditions is met). Determine the probability for the first player to win the game.
4. Two dice are rolled simultaneously. What is the probability that the sum is even given that it is a multiple of 3?
5. There are 5 white balls and 7 green balls in the first urn; 2 white balls and 10 green balls in the second urn. The third urn, that has initially been empty, is filled with the balls: 4 balls are taken from the first urn, 6 balls are taken from the second urn, and they are placed into the third urn. After that, 2 balls are taken at random from the third urn. It turns out that both these balls are green. Determine the probability that these balls originate from different urns.

Section 2

1. Is it possible for random variable ${\textstyle X}$ to have a binomial distribution if (a) ${\textstyle EX=6}$ and ${\textstyle VarX=3}$; (b) ${\textstyle EX=7}$ and ${\textstyle VarX=4}$?
2. Let ${\textstyle Y}$ be number of sixes and ${\textstyle Z}$ be number of fours one gets when rolling six dice. Find the expected value and variance of ${\textstyle Y+Z}$.
3. Let ${\textstyle Z}$ be a random variable with geometric distribution. Prove that ${\textstyle P(Z=n+k|z>n)=P(Z=k)}$ (lack of memory property of geometric distribution).
4. Let us consider a sphere of radius ${\textstyle R}$ centered at ${\textstyle O}$. Point ${\textstyle M}$ is chosen at random inside this circle. Random variable ${\textstyle \xi }$ is equal to the length of ${\textstyle OM}$. Find the cumulative distribution function, probability density, expected value and variance of ${\textstyle \xi }$.
5. Random variable ${\textstyle \theta }$ is exponentially distributed with parameter ${\textstyle \lambda }$. Calculate the probabilities that ${\textstyle \theta }$ belongs to intervals ${\textstyle (0;1),(1;2),\ldots ,(n-1;n),\ldots }$ and show that these probabilities form a geometric sequence. What is the common ratio of this sequence?
6. It is known that ${\textstyle \xi }$ is normally distributed random variable, and ${\textstyle P\{|\xi -E\xi |<1\}=0{.}3}$. Find the probability that ${\textstyle |\xi -E\xi |<2}$.

Section 3

1. The probabilities for three students to pass the exam are equal to ${\textstyle {\frac {11}{12}}}$, ${\textstyle {\frac {11}{14}}}$ and ${\textstyle {\frac {18}{25}}}$ respectively. Determine the probability that at least one student passes the exam given that they pass or fail independently of each other.
2. Let us consider independent identically distributed random variables with uniform distribution on ${\textstyle (\theta ;\theta +3)}$. Find the maximum likelihood estimator of ${\textstyle \theta }$. Which one of these estimators is unbiased? Justify your answer.
3. Find the smallest possible value of ${\textstyle P{\bigl (}|\xi -E\xi |\leq 3{\sqrt {Var\xi }}{\bigr )}}$.
4. The probability that a new-born baby is a boy is equal to ${\textstyle 0{.}52}$. Find the interval which contains the quantity of boys out of ${\textstyle 10\,000}$ newborn babies with probability ${\textstyle 0{.}98}$.
5. Prove that for multivariate normal distribution uncorrelatedness implies independence.
6. Use characteristic functions to show that a sum of independent (and not necessarily identically distributed) random variables also has normal distribution.

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.