Chapter 2 Section 1 Question 10
Question
Let X be a random varriable
  1. It is necessarily true that $X\geq 0$?
  2. Is it necessarily true that there is some real number c such that $X+c \geq 0$?
  3. Suppose the sample space S is finite. Then is it necessarily true that there is some real number c such that $X+c\geq 0$?
  4. Compute $Y(4)$
Answer
Author: Mohammad-Ali Bandzar| Date:Oct 14 2020
(a) It is necessarily true that $X\geq 0$?
No, X can be any real number
(b) Is it necessarily true that there is some real number c such that $X+c \geq 0$?
No, for example let $S=\{1,2,3,4...\}$ and let $X(s)=-s$. No c will exist such that $X+c\geq 0$ as an s can always be chosen to make $X+c\leq 0$
(c) Suppose the sample space S is finite. Then is it necessarily true that there is some real number c such that $X+c\geq 0$?
Yes, in a finite sample space, if we set $c=min_{s\in S}X(s)$ then we would have $X+c\geq 0$