1
$\begingroup$

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.

enter image description here

Formal Definition 2.

enter image description here

Why are there two types of definitions in existence?

$\endgroup$
0

1 Answer 1

3
$\begingroup$

The two definitions are the same.

  • 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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.