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

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:


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from scipy.stats import expon

# Parameters of the exponential distribution
mean_waiting_time = 5

# Calculate that new customer will arrive in less than 3 minutes
probability = expon.cdf(3, scale=mean_waiting_time) - expon.cdf(0, scale=mean_waiting_time)

# Print the result
print(f'The probability that the next customer arrives within 3 minutes is: {probability:.4f}')

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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Section 4. Chapter 2