Digital Signal Processing (DSP) with Python Programming

1Useful Maths

1.1. Basic concepts on probability

Without describing in detail the formalism of the probability theory, we simply remind the reader of useful concepts. However, we advise the reader to consult some of the many books with authority on the subject [BIL 12].

In probability theory, we consider a sample space Ω, which is the set of all possible outcomes ω, and a collection of its subsets with a structure of σ-algebra, the elements of which are called the events.

DEFINITION 1.1 (Random variable).– A real random variable X is a (measurable) application from the Ω to :

[1.1]

DEFINITION 1.2 (Discrete random variable).– A random variable X is said to be discrete if it takes its values in a subset of , at the most countable. If {a₀, …, a_n, …}, where n ∈ , denotes this set of values, the probability distribution of X is characterized by the sequence:

[1.2]

representing the probability ...