Course Content

Probability Theory Basics

1. Basic Concepts of Probability Theory

Stochastic Experiment and Random Event Probability and It's Properties Geometrical Probability Challenge: Solving the Task Using Geometric Probability Independence and Incompatibility of Random Events Conditional Probability

2. Probability of Complex Events

Inclusion-Exclusion Principle Challenge: Solving the Task Using Inclusion-Exclusion Principle The Multiplication Rule of Probability Law of Total Probability Bayes' Theorem Challenge: Solving the Task Using Bayes' Theorem

3. Commonly Used Discrete Distributions

Binomial Distribution Challenge: Solving Task Using Binomial Distribution Multinomial Distribution Geometric Distribution Poisson Distribution Challenge: Solving Task Using Poisson Distribution

4. Commonly Used Continuous Distributions

Continuous Uniform Distribution Exponential Distribution Challenge: Solving Task Using Exponential Distribution Gaussian Distribution Challenge: Solving Task Using Gaussian Distribution

5. Covariance and Correlation

What is Covariance?What is Correlation?Challenge: Solving the Task Using Correlation

Gaussian Distribution

The Gaussian distribution, also known as the normal distribution, is a continuous probability distribution widely used in statistics and probability theory.

Gaussian distribution applications

We can use this distribution to describe the following values:

Physical Measurements: Many physical measurements, such as height, weight, blood pressure, and body temperature, can be reasonably approximated by a Gaussian distribution. For example, the height of adult men or women in a population often follows a Gaussian distribution;
Errors and Residuals: In statistical analysis or regression modeling, errors or residuals (the difference between observed and predicted values) are usually assumed to be normally distributed;
Test Scores: Standardized test scores such as the SAT or ACT are often modeled using a Gaussian distribution in educational testing;
Environmental measurements: A Gaussian distribution can often describe variables such as air pollution, noise levels, and water quality measurements.

Python implementation

We can also use .cdf() method of scipy.stats.norm class to work with Gaussian distribution in Python. It has two main parameters: loc determines the mean value of the experiment's result, and scale determines the average deviation from the mean.

Example

Calculate the probability that the height of a randomly chosen man will be less than 160 or more than 190. Assume that the mean value of men's height is 170, and the average deviation is 20.


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from scipy.stats import norm

# Calculate probability that man has height between 160 and 190
proba_160_190 = norm.cdf(190, loc=170, scale=20) - norm.cdf(160, loc=170, scale=20)

# Calculate the probability of complemented event
prob_not_160_190 = 1 - proba_160_190
print(f'Result probability is {prob_not_160_190:.4f}')

The Gaussian distribution is one of the most popular and commonly used distributions. Its properties are discussed in more detail in the Probability Theory Mastering course.

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Section 4. Chapter 4