Probability Final: Is (a) or (b) a Valid Random Variable?

[Shackleford]

I'm reviewing the early stuff, and I'm a bit fuzzy on what is a valid random variable. I don't see why (b) is valid while (a) is not. For (b), it says the set is in the family of subsets for all x. There's no mapping from the sample space that is less than or equal to -5. Well, then I thought it might be the null set satisfies this. However, why doesn't the null for (a) satisfy X(s) ≤ 1?

http://i111.photobucket.com/albums/n149/camarolt4z28/Untitled-7.png?t=1304620770

 

[snipez90]

First of all, asking whether the null set satisfies some condition is vacuous.

{s in S | X(s) ≤ 1} = {1} which indeed is not in E.

 

[Shackleford]

Do you understand my question? Certainly, I see there is no mapping from the sample space that satisfies that. However, it has to be valid for all x to be a random variable. How does (b) satisfy that outside of the sample space mapping? For example, for {s in S | X(s) ≤ -5} = {?}

 

[snipez90]

I understand the actual problem. Still not sure what you're misunderstanding.

In part b), {s in S | X(s) ≤ x} is the empty set for any x in (-infinity, -1), and the empty set is in E, so yes obviously {s in S | X(s) ≤ -5} = empty set, which is in E.

For part a), X(1) is less than OR EQUAL TO 1, so {s in S | X(s) ≤ 1} = {1}, which is NOT in E.