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

Poisson Distribution

Assume that we have some sequence of events that occurred in some period of time with the following properties:

Events are independent;
The simultaneous occurrence of two or more events has low probability (in this case, simultaneity is implied in the context of the occurrence of events in an extremely small time interval - up to seconds);
Probabilistic characteristics of the occurrence of an event do not depend on time.

In this case, this set of events is called Poisson point process.

Poisson point process examples

The examples of Poisson point processes are:

the arrival of cosmic particles on the counter;
client requests to the server on a certain day of the week;
road accidents on a certain section of the road on a certain day;
insurance cases with clients of a certain insurance company.

Note
It is important to understand the difference between the Bernoulli and Poisson processes. In the case of the Bernoulli process, we independently experiment and count the number of successes.
At the same time, the Poisson process describes events in nature that we do not directly influence but only observe their appearance.

Poisson distribution

The Poisson distribution is a discrete probability distribution representing the number of events occurring in a fixed time interval in the Poisson point process.
This distribution has one parameter representing the average number of events occurring in a one-time unit.

Task example

Let's solve the following task using Poisson distribution:

In a call center, calls are received at an average rate of 5 calls per minute. What is the probability of receiving from 290 to 310 calls in an hour?


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from scipy.stats import poisson

# Parameters
calls_per_minute = 5  # Average rate of calls per minute
calls_per_hour = calls_per_minute * 60

# Calculate the probability using the Poisson distribution
# We will add probabilities of occurring 290, 291, ..., 310 calls
# We can simply add probabilities because events of occuring 290, 291,.. calls
# are incompatible
probability = 0
for i in range (290, 311):
	probability = probability+poisson.pmf(i, mu=calls_per_hour)

# Print the results
print(f'Corresponding probability is: {probability:.4f}')

In the code above, we used .pmf() method of scipy.stats.poisson class to calculate probability in each of the points 290, 291, ... , 310 and summarized al these probabilities to calculate final result.
Parameter mu determines the average number of accidents over a period of time.
If you want to calculate the probability for a different period of time, then in the mu parameter, specify the average number of events in the desired period.

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Section 3. Chapter 5