Intuition behind Random Variables in Probability Theory

Math

Anyone who has taken a probability theory course has encountered the concept of Random Variables. Random variables are of vital importance in developing a more profound understanding of the world of probabilities and all the interesting results that it entails.

Before we dive into the intuition behind random variables let’s do a quick recap of the core ideas and concepts in probability theory. The following is not meant to be an introduction to the subject. We assume that the reader has already encountered all these concepts and we merely hope to refresh them a bit. Moreover, we do not aim to analyze all the related concepts such as distributions or mean and variance. We simple hope that by the end of the article the reader will have understood the reason why random variables are essential.

A set is a collection of objects without repetition and without ordering. {a, 6, “Panos”, red, / } is an example of a set.

We can define a set in three distinct ways:

There are numerous operations we can perform upon sets. Two of the more frequently-encountered operations are the ones of union and intersection.

The union of two sets is a set containing all elements that are in A or in B (could be both). For example, {1,2,3} ∪ {2,3,4}={1,2,3,4}. Thus, we can write x ∈(A∪B) if and only if (x∈A) or (x∈B).

The intersection of two sets A and B, denoted by A∩B, consists of all elements that are both in A and B. For example, {1,2}∩{2,3}={2}.

Finally, we define the powerset of a set A as the set of all A’s subsets including the empty set. For example, the powerset of the set {1,2,3} is the set {{∅}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}. In this article we will use the following notation, S(A), to denote the powerset of a set A.

A probability model is a mathematical description of an uncertain situation.
Every probability model refers to a process called experiment which produces exactly one…

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