Probability and Statistics
Spring 2019
Course Description
Description: This course closes the sequence of mathematical courses offered at Inha University in Tashkent. In this course we emphasize and illustrate the use of probabilistic models and statistical methodology that is employed in countless applications in all areas of science and engineering.
Prerequisite(s): It is necessary to be familiar with the concepts of Calculus to learn elementary probability theory and properties of distribution of random variables. In additional, a modest amount of a matrix algebra is used to support the linear regression models, Markov chains and Stochastic simulations.
Prerequisite courses: Calculus 1 (Mandatory), Linear Algebra (Mandatory)
Attendance: It is mandatory to attend at least 20 lectures. The attendance lists are updated manually every week.
Course Format
Study materials: Students are recommended to read the Lecture Notes and Study Guides before and after the lectures. It is necessary to test their knowledge using the online quizzes designed at this web-page.
Homework and Quiz: There are also 2 (two) Mandatory Homework assignments during the semester. We assign 10 (ten) short-quizzes during the lectures with similar questions from homework assignments and online quizzes. Students performed the homework and the online quizzes should be able to succeed the quizzes.
2 (two) worst quiz and homework results will be dropped to allow students to be absent.
Software projects: There are four software projects assigned with different level of difficulties. Students should submit them in a group of two students personally during the assigned slots. The list of software projects:
- Creating random variables with Negative Binomial Distribution;
- Probabilities of Radom variables with Normal Distribution;
- Experimental probability and simulations;
- Stochastic simulations and Markov chains;
Week 1: Introduction. Combinatorial Analysis.
- Classical rule of calculating probabilities.
- Fundamental counting rule.
- Permutations.
- Combinations.
Lecture 2.2 | Conditional Probability.
- Conditional probability.
- Law of total probabilities.
- Bayes' theorem.
Lecture 3.2: Random Variables.
- Discrete & Continuous variables.
- Mass function.
- Properties of distribution functions.
Lecture 4.1: Discrete random variables.
- Probability distributions.
- Cumulative distribution.
Lecture 4.2: Expected Value & Variation.
- Expected value. Properties.
- Variation. Properties.
Lecture 5.1: Binomial Distribution
- Bernoulli trial
- Binomial experiment
- Properties of the binomial distribution
Lecture 5.2: Discrete distributions
- Poisson distribution
- Geometric distribution
- Negative binomial distribution
Lecture 6.1: Continuous Random Variables
- Density function
- Properties of the density function
Lecture 6.2: Expected Value & Variance
- Examples of density function
- Cumulative distribution function
- Expected Value & Variance
- Conditional Probability in Continuous Case
Lecture 7.1: Intro to continuous distributions
- Uniform distribution
- Intro to Normal distribution
Lecture 7.2: Continuous distributions
- Standard normal distribution
- Finding probabilities for normal distribution
Lecture 8.1: Exponential distribution
- Intro to Exponential distribution
- Exponential vs Poisson
- Expected value & variance
- Memoryless distributions
Lecture 9: Limit Theorems
- Generating random values
- the Law of Large Numbers
- Central Limit Theorem
Lecture 10.1: Joint distributions.
- Joint distributions.
- Marginal distributions.
- Independent variables.
- Covariance.
Lecture 10.2: Distribution of sum of variables
- Convolution of distributions
- Density function of sum of two variables
- Expected values of sum of two variables
Lecture 11.1: Introduction to Statistics
- types of Statistics: Inferential & Descriptive
- sampling methods
- levels of data measurements
- histograms
Lecture 11.2: Types of Statistics Graphics
- Histograms, polygons, bar graph
- Dot plot, Stem plot, Scatter plot
- Pareto chart, Pie chart
- Examples of bad graphics
Lecture 11.3: Measures of center and variance
- Measures of center: mean, median, mode
- Measures of variance: standard deviation
- Chebyshev's theorem on proportions
Lecture 12.1: One point parameter estimation
- Inferences about population parameter
- Likelihood function, log-likelihood function
- Maximum likelihood estimates
- Likelihood functions in continuous case
Lecture 12.2: Confidence intervals for mean
- level of confidence, critical values
- error margin
- confidence intervals for mean (big-samples)
Lecture 13.1: Confidence intervals | continued
- t-distribution
- confidence intervals for mean (small-samples)
- confidence intervals for variance
- chi-squared distribution
Lecture 13.2 | Hypothesis Test. P-Values.
- null hypothesis & alternative hypothesis
- p-value: plausibility of the null hypothesis
- left-tailed, right-tailed, two-tailed
- testing hypothesis about population mean
Lecture 14.1 | Hypothesis Test for proportions, variance
Lecture 14.2: Final Review.