Random Variables: Discrete, Continuous & Probability Theory

Random Variables and Probability Concepts

Random variable.
Meaning: So far we have studied general characteristics of probability spaces (Ω, F, P), where Ω is the sample space, F is a σ-algebra and P is a probability function. Often we are not interested in studying the events of the σ-algebra themselves, but one or several numerical characteristics associated with the outcome of the experiment. This may be, for example, the event of selecting a student from the university, the probability that your grade is greater than or equal to 5 (likely to pass), or when establishing a new tax the probability that an individual's annual income does not exceed 20,000 euros. These numerical characteristics, which under certain conditions we can associate with a probability induced by the probability of the events of the experiment, are called random variables.

Definition 1: Random Variable

Given a random phenomenon associated with a probability space (Ω, F, P), a random variable X is a function defined on the sample space Ω whose values are real numbers and which verifies the measurability condition: for every x ∈ ℝ the set {ω ∈ Ω | X(ω) ≤ x} ∈ F.

Definition 2: Alternative Formulation

This in turn allows us to define a random variable as any function defined on the sample space that takes values in the real numbers such that the inverse image of each interval of the form (-∞, x] is an event in the σ-algebra F (i.e., measurable).

1.1.1 Random Variable Types

Within the set of random variables we distinguish two main types in order to work with them more simply, knowing the group they belong to. Basically there are two types of random variables: the discrete type and the continuous type. Although we could add a third group where there are those that belong neither to one nor the other — these are called mixed and are not discussed here — most treatments focus on discrete and continuous variables.

Thus we can describe in a simple way the probabilities of events using a new function: the density function (for continuous variables) or the probability mass function (for discrete variables).

1.1.1.1 Discrete-Type Variable

A random variable X is said to be of discrete type if the set of values that this random variable takes, X(Ω), is a countable set (finite or countably infinite). When all the possible outcomes of an experiment form a finite or countable set, the associated random variable is of discrete type. Examples include drawing a winning combination in a lottery, the number of telephone calls a call center receives in a day, the number of successes in a sequence of Bernoulli trials, etc.

Continuous-Type Random Variables

There are random phenomena that fit the continuous model. Consider, for example, the time it takes for a person to be served in a queue. In this case the set of possible outcomes is an uncountable set of numbers in an interval such as [0, +∞), and the probability that the service time equals any single exact value is typically zero. Instead, the probability of being served before time x is given by a continuous cumulative distribution function that increases smoothly with x. Small increments of x produce small increases in probability. For continuous random variables we often describe their behavior using a probability density function (pdf) from which probabilities of intervals are obtained via integration.

Note: A mixed random variable has both discrete and continuous components; such variables are not treated in detail here but are an important extension in probability theory.

Examples recap:

• Discrete: number of calls to a call center, count of defective items, lottery draw outcomes.
• Continuous: waiting times, lifetimes, measurement errors, and many physical quantities.

These distinctions help select appropriate tools: probability mass functions (pmf) and cumulative distribution functions (cdf) for discrete variables; pdfs and cdfs plus integration for continuous variables; and a combination of both for mixed variables.