A discrete random variable has a countable number of states. It can take on only certain values along a given interval. This variable can either be an integer or a named state that does not have a numerical value. Discrete data is counted.

Discrete Random Variable

A continuous random variable can take on any value at any point along a given interval. This variable is a real number. Continuous data is measured.

Continuous Random Variable

- Discrete Random Variable
- Continous Random Variable

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A variable that holds different values by chance as a result of a random experiment.

Random Variable

Goodfellow, I., Bengio, Y., & Courville, A. (2016). $\mathit{Deep \ Learning.}$ MIT Press. Retrieved from [www.deeplearningbook.org](https://www.deeplearningbook.org) 

Deep Learning

Random variables are typically denoted using a plain typeface lowercase letter and the values it can take on with lowercase script letters.
- The random variable a can have the values of a1 and a2

For vector-valued variables:
- The random variable x can have one of its value as 𝑥

Random Variable Denotation

Sampling is the formal process of acquiring data by drawing independent realizations from an underlying random process. Collecting this empirical data is the necessary first step in using statistical methods to investigate and estimate unknown theoretical probabilities.

Sampling from a Random Process

In probability theory, an event represents a subset of the underlying sample space. When working with a random variable $$X$$, an occurrence where the variable takes on a specific value $$v$$ (denoted by $$X=v$$) or falls within a specific range of values (such as $$1 \leq X \leq 3$$) is considered an event. A probability score, denoted like $$P(X=v)$$ or $$P(1 \leq X \leq 3)$$, is assigned to these events to represent their likelihood.

Probability Event

Every value taken by a random variable corresponds to a subset of the underlying sample space. Thus, the occurrence where a random variable $$X$$ takes a specific value $$v$$, denoted by $$X=v$$, is considered an event. Its probability is denoted as $$P(X=v)$$. This concept also extends to ranges, such as $$P(1 \leq X \leq 3)$$, representing the probability that $$X$$ falls within that range.

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