Probability Theory Basics

1. Basic Concepts of Probability Theory

We will start our way of learning probability theory by considering some basic definitions and rules: what is a stochastic experiment and random event, what is independence and incompatibility of events in the context of probability theory, what is the probability and how can we calculate probabilities of different elementary events.

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

In real-life tasks, we often have to deal with complex relationships and, as a result, calculate probabilities of several events or events that depend on each other. Let's consider how we can do this using probability theory.

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

To solve many real problems in probability theory, special models have been created that describe a particular situation. Let's consider some of the most used models that can be used to describe some discrete results of stochastic experiments.

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

What if the result of a stochastic experiment cannot be described by a discrete value? For this, models that work with continuous values are used. Consider the most popular of these models.

Continuous Uniform Distribution

Exponential Distribution

Challenge: Solving Task Using Exponential Distribution

Gaussian Distribution

Challenge: Solving Task Using Gaussian Distribution

5. Covariance and Correlation

Often we are faced with the task of checking the dependence of the results of different stochastic experiments on each other. Moreover, it is necessary not only to assess the presence of dependencies but also to somehow quantify the degree of dependencies. To solve these problems, we can use covariance and correlation.

What is Covariance?

What is Correlation?

Challenge: Solving the Task Using Correlation

Exponential Distribution

The exponential distribution is a continuous probability distribution that models the time between events in a Poisson process.

We remember that the Poisson process describes the number of events that occurred during some period. On the other hand, exponential distribution describes the time between the occurrence of two successive events (the distance between two adjacent non-zero points in the graph above).
Poisson distribution is parameterized by the mu parameter, which describes the average number of accidents by the time unit. The exponential distribution is parametrized by the parameter scale which determines the average time between two accidents.

Note
There is a clear relationship between the Poisson process parameter mu and the exponential distribution parameter scale:
mu = 1 \ scale for one unit of time

Example

Suppose the average time between customer arrivals at a store is 5 minutes. What is the probability that the next customer arrives within 3 minutes?

We have a Poisson process where an event is the customer's arrival. The average time between two arrivals is 5 minutes. As a result, we can use exponential distribution to calculate the corresponding probability:

We also use .cdf() method of the scipy.stats.expon class with a specified scale parameter to calculate the corresponding probability on the interval [0,3].

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Зміст курсу

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

Probability Theory Basics

Exponential Distribution

The exponential distribution is a continuous probability distribution that models the time between events in a Poisson process.

Note
There is a clear relationship between the Poisson process parameter mu and the exponential distribution parameter scale:
mu = 1 \ scale for one unit of time

Example

Suppose the average time between customer arrivals at a store is 5 minutes. What is the probability that the next customer arrives within 3 minutes?

We also use .cdf() method of the scipy.stats.expon class with a specified scale parameter to calculate the corresponding probability on the interval [0,3].

Все було зрозуміло?

Секція 4. Розділ 2