Course Content

Advanced Probability Theory

1. Additional Statements From The Probability Theory

Course Overview Absolutely Continuous and Discrete Random Variables Cumulative Distribution Functions and Probability Density Functions Characteristics of Random Variables Random Vectors Useful Properties of the Gaussian Distribution Challenge: Detecting Outliers Using 3-Sigma Rule

2. The Limit Theorems of Probability Theory

Law of Large Numbers Law of Large Numbers for Bernoulli Process Challenge: Estimate Mean Value Using Law of Large Numbers Central Limit Theorem Challenge: Application of the CLT to Solving Real Problem

3. Estimation of Population Parameters

General population. Samples. Population parameters.Momentum estimation. Maximum Likelihood Estimation Challenge: Estimate Parameters of Chi-square Distribution Unbiased Estimation Challenge: Checking Bias of An Estimation Using Simulation Consistent Estimation Efficient Estimation Confidence Intervals for Population Parameters Challenge: Confidence Interval for Exponential Distribution Parameter

4. Testing of Statistical Hypotheses

What is Statistic Hypothesis? Type 1 and Type 2 Errors What is P-value?Comparing Means of Two Different Datasets Challenge: Using CLT to Compare Mean Values of Non-Gaussian Datasets Challenge: Resampling Approach to Compare Mean Values of the Datasets Testing the Hypothesis of Independence of Two Random Variables

Course Overview

This course is a logical continuation of the Probability Theory Basics and Learning Statistics with Python courses. Its aim is to provide a comprehensive understanding of advanced concepts that can be applied to solve real-life tasks for data analytics and data science positions.

Course structure

The first section covers necessary additional topics not addressed in the Probability Theory Basics course. Here, you will gain an understanding of random variables, their probability distributions, and the characteristics used to describe them. Additionally, you will learn about specific features of the Gaussian distribution, making it widely popular and used;
The second section delves into the limit theorems of probability theory. These theorems form the basis for statistical inference and hypothesis testing, widely employed in solving real-life tasks;
In the third section, we will explore how to accurately estimate unknown parameters of probability distributions from a statistical perspective. Correct estimation is crucial to ensure that our results accurately depict real-life processes we aim to model and analyze;
The final section focuses on testing statistical hypotheses. You will learn about hypotheses, their role in data analysis, and how to test them correctly to obtain informative results.

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Section 1. Chapter 1