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

Continuous Uniform Distribution

Continuous distribution describes the stochastic experiment with infinite possible outcomes.

Continuous uniform distribution

Continuous uniform distribution describes an experiment where all outcomes within the interval have an equal probability of occurring.
If the variable is uniformly distributed, we can use a geometrical approach to calculate probabilities.

Example

Consider a line segment of length 10 units. What is the probability of randomly selecting a point on the line segment such that the distance from the starting point to this point is between 3 and 7 units?

As a result, the position or the point is uniformly distributed on the line with length 10.
We can simply divide the length of the desired interval by the whole length of the segment.
We can also use the .cdf() method on the scipy.stats.uniform class to calculate the corresponding probability:


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from scipy.stats import uniform
# Parameters of an experiment
whole_length = 10
range_start = 3
range_end = 7

# Geometrical approach
desired_length = range_end - range_start
geom_proba = desired_length / whole_length
print(f'Geometrical probability is {geom_proba:.4f}')

# Using `.cdf()` method
upper_cdf = uniform.cdf(range_end, loc=0, scale=whole_length)
lower_cdf = uniform.cdf(range_start, loc=0, scale=whole_length)
cdf_proba = upper_cdf - lower_cdf
print(f'Probability using .cdf() is {cdf_proba:.4f}')

The first parameter of the .cdf() method determines the point at which we calculate probability; loc parameter determines the beginning of the segment, and scale determines the length of the segment.

The .cdf() method calculates the probability that an experiment's result falls into a certain interval: .cdf(interval_end) - .cdf(interval_start).
We will consider this method in more detail in Probability Theory Mastering course.

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Section 4. Chapter 1