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Suppose that we have an expression of random variables including $X-X$ or $2X-X$ or $XY-XY$ and so on. can we treat random variables as real numbers? That is, can we delete $X-X$ or replace $2X-X$ by $X$? what about $X/X$?

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If $X-X$ stands to denote $X(\omega) - X(\omega)$ then, yes, it is zero, but if you meant $X_1-X_2$ were $X_1$ and $X_2$ are identically distributed, but independent random variables, than $X_1-X_2$ need not be zero. –  Sasha Apr 1 '13 at 18:06
    
X−X stands to denote X(ω)−X(ω) , they are the same random variable. –  May Apr 1 '13 at 18:17

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up vote 3 down vote accepted

Random variables are functions. Any algebraic step valid for functions is therefore valid for random variables. In particular, we can replace $2X-X$ by $X$, and, with the usual caution about division by $0$, we can replace $\frac{X}{X}$ by $1$.

Remark: Imagine tossing a fair die twice. The result on the first toss is a random variable. The result on the second toss is a random variable. These two random variables have the same distribution. But they are not the same random variable. Calling both of them $X$, and manipulating "algebraically" will lead to disaster.

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@Nicolas: Thank you a lot. You answer is really helpful. Just a question about the Remark; In my mind, Tossing a fair dice twice would result in different outcomes (not different random variables), but tossing a dice itself is a random variable. is it correct? –  May Apr 1 '13 at 18:28
    
They are different random variables. The range of the two functions is the same, but the domains are not. –  André Nicolas Apr 1 '13 at 18:37

Formally, a random variable $X$ is a mapping $X:\Omega\to\mathbb{R}$ where $\Omega$ is the set of outcomes (sample space) in a probability space. The mapping also needs to be measurable but that is less important here.

So if $X$ and $Y$ are two random variables, then how can $X-Y$ be a random variable? The answer is that it is defined as the mapping $(X-Y):\Omega\to\mathbb{R}$ which sends an outcome $\omega$ to the difference of the real numbers $X(\omega)$ and $Y(\omega)$, that is, it is defined by:

$$ (X-Y)(\omega) = X(\omega)-Y(\omega) $$

The minus on the right-hand side is a minus between two real numbers. So this is "pointwise subtraction". It entirely the same as $f-g$ for two functions $f$ and $g$.

So since for every $\omega$, $X(\omega) - X(\omega) = 0$ (zero the number), it is true that the mapping $X-X$ is the mapping that sends everything to zero, that is the zero function $0$. So $X-X=0$ (zero the function).

So the answer is yes.

Regarding $X/X$: In the points where $X(\omega)$ is not zero, this is the constant function $1$. But in the points where $X(\omega)$ is zero, this is not defined ($0/0$). So if you know that $X$ is never zero, it's safe to replace $X/X$ by $1$. If you know $X$ is zero in some points $\omega$, formally $X/X$ is not defined on all of $\Omega$ and so is not a random variable.

(However, in the last case, the event $\{X=0\} = X^{-1}(\{0\})$ may have zero probability even if it's non-empty as a set, and in that case it's not a problem in practice to disregard the division by zero, since the random variables are "identified" if they only disagree on a null set (null event).)

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