Probability & Statistics Ch.2 Section 1 Question 10

Solutions for “Probability and Statistics: The Science of Uncertainty” (Second Edition). These are solutions I have come up with; I offer no guarantee of accuracy.

Question

Let X be a random variable.

a) Is it necessarily true that $X \geq 0$? b) Is it necessarily true that there is some real number c such that $X + c \geq 0$? c) Suppose the sample space S is finite. Then is it necessarily true that there is some real number c such that $X + c \geq 0$?

Solution

(a) Is it 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 $X + c \geq 0$ because 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 $X + c \geq 0$.