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

Absolutely Continuous and Discrete Random Variables

Understanding Random Variables

A random variable is a value that changes based on the outcome of a random event or experiment. For example, it could be the number of heads in coin tosses, accidents on a road segment, weight on a scale, or bus wait time.

Random variables can be categorized into two types:

Discrete Variables: Take specific, countable values like natural numbers (e.g., the number of road incidents);
Continuous Variables: Can take any value within a certain range (e.g., a person's weight).

Probability Mass Function (PMF)

Probability Mass Function (PMF) is used to describe discrete random variables. It's like a table containing all possible values of the random variable and their corresponding probabilities.

For example, consider a simple PMF describing the outcome of a single coin toss.

Let's look at the example of using PMF in Python: we will use the Binomial distribution described in the Probability Theory Basics course.


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from scipy.stats import binom

# Flipping a coin 5 times and finding out how many times we get heads
# `n` is a number of experiments, `p` is probability of occuring a head
coin = binom(n=5, p=0.5)

# Due to conditions of an experiment we can have heads 0, 1, 2, 3, 4, 5 times
# Using pmf to get probabilities of all described results of an experiment
for i in range (6):
	print('Probability of ', i, ' heads is ', round(coin.pmf(i), 4))

The result of the code above can be represented as follows:

Note

The sum of all probabilities in the PMF table always equals 1.

On the other hand, how can we represent continuous random variables if we cannot simply list all possible values of these variables and write down the distribution series? We will address this question in the next chapter.

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Section 1. Chapter 2