# What is the difference between the following two definitions of a Random Variable?

Please, explain in layman's terms, as far as possible.

What is the difference between the following two definitions of a Random Variable?

Formal Definition 1.

Formal Definition 2.

Why are there two types of definitions in existence?

• In Definition 1, the set $\{s\colon X(s) \in \mathcal I\}$ is a subset of the sample space, and the definition says that it must be that this subset is an event (that is, a member of the (unmentioned) event field $\mathcal F$) for all possible choices of interval $\mathcal I$ of the real line that we might make. It follows that the inverse image of each Borel set of the real line is an event in $\mathcal F$. Note that $X$ is a single or univariate random variable.
• Definition 2 muddies the waters by considering $X$ to be a random vector or $n$-dimensional or $n$-variate random variable, but taking $n = 1$, we see that the definition is saying that the inverse image of each Borel set is an event in $\mathcal F$ (cf. the last sentence). In fact, as that last sentence adds, it suffices to insist that the inverse images of the intervals are events in $\mathcal F$, which is what Definition 1 says.