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I was going over notes today on properties and proofs of random variables, in particular the proof that $E(cX_{n}) = \lim_{n \to \infty}E(cX_{n})$.

In the definition, it states that $X$ is a non-negative random-variable and this is denoted as $0 \leq X$. Also that $c$ is an element of the positive reals. This is denoted as $c \in \mathbb{R}^{+}$.

I was curious as to why there is a difference in notation, if both $c$ and $X$ are positive, then why are they written differently? And hence is there a difference in meaning between the two or is just down to personal choice?

Hamish Gibson
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The expression "$X \ge 0$" does not mean "$X$ is positive". Instead, it means that every value of $X$ is either positive or zero.

If you really wanted notation for "$X$ is positive", then you could write $X > 0$, and it is indeed your own preference.

Keep in mind that there is a subtle distinction between $c>0$ and $X>0$. First, $c$ denotes a constant, having only one value, and so the expression $c>0$ means that this one value is positive. On the other hand $X$ is a random variable, having many possibly different values each time you sample it, and the expression $X > 0$ means that each time you sample it the value will be positive.

In fact there usually are multiple ways to express the same mathematical thought, just as there are usually multiple ways to express the same thought in any branch of human discourse, and the choice of expression is in general, as you say, a matter of personal preference, tempered by context, or clarity, or a number of other factors.

Lee Mosher
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