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Given some random variable $X$, how should I interpret addition: $X+X$?

Two independent potentially different results added together? Or on the other hand, one result doubled $X+X= 2X$.

Similarly, is $X-X$ equal to $0$?

References:

David
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    Related question: https://math.stackexchange.com/questions/1618991/is-the-sum-of-random-variables-x-and-x-2x – David Nov 02 '22 at 05:06
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    Hi, just saying, you can also use MathJax (math typeset) in your question heading. – Cheese Cake Nov 02 '22 at 05:06
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    Arithmetic does not go away because the quantities are measurable functions. – copper.hat Nov 02 '22 at 05:10
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    $X + X$ is just $2X$ and $X - X$ is just $0$ as usual. If you want to sum two independent copies of $X$ you should refer to them using different notation, e.g. $X_1 + X_2$, because they are not literally the same random variable. – Qiaochu Yuan Nov 02 '22 at 05:54
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    Interesting there seems to be a difference of opinion floating around: https://stats.stackexchange.com/a/235685/100648 – David Nov 02 '22 at 16:56
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    @David I feel like what you should be getting out of the link you just cited is that among mathematicians there is no difference of opinion about $X+X$: all are in agreement that it is the same as $2X$. What AP Central thinks about notation for random variables is another matter. (But I worry that we may be approaching the $6/3(-2)$ levels of nonsense here: the point is that the sum of a random variable with itself is different from the sum of two independent copies and you always make sure to clearly communicate which you mean. Everything else is commentary.) – Misha Lavrov Nov 03 '22 at 00:13

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