Q: Let X be a random variable.
a) Is it necessarily true that there is some real number c such that X + c >= 0?
b) Is the above true if the sample space is finite?
I believe that a) is false and b) is true. The question is basically asking whether or not we can find a bound on X. That is to say is there a c where X(z)>=c for all z in the sample space.
For a), my idea was to state that since X is infinite, there is not necessarily a minimum value to X. Therefore it becomes impossible to find a c such that c<=X(z) for all z because c-1 might belong to X but c-1<c.
For b), my idea was to take any finite set and state that because the set is finite, there must be a minimum value to X. Let us assume this minimum value occurs at X(z). If we let c=X(z), then it becomes true that for w =/= z, X(w)>=X(z)>=c.
Any thoughts or suggestions would be greatly appreciated.