Random Variables

Lesson 4 Chapter 2

In probability theory, the outcomes of a random phenomenon determine the random variable values. In other words, a random variable is a variable that its values are determined with a random event. We should be able to measure a random variable that provides the capability to assign probabilities to its possible values. The domain of a random variable is the sample space. For example, in the case of having a dice, only six possible outcomes are considered, as {1,2,3,4,5,6}.

Notation: In mathematics, a random variable is usually denoted with upper case roman letters such as $X$ , $Y$ . However, such notation is not consistent, and you should expect to see different notations. However, don't worry, the utilized notation would be usually reported whenever you read any article.

By using more precise mathematics notation, a random variable $X$ is a measurable function defined as $X: \Omega \rightarrow E$ , which is from all possible space $\Omega$ to some event $E$ . Let's have an illustrative example. Assume we would like to roll a dice, and the measurable event space is all numbers less than 4. Here, we have $\Omega=\{1,2,3,4,5,6\}$ , the random variable $X$ as the outcome of rolling the dice, and we define the event as $E \equiv X \leq 4$ .

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