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Is it true that $|X|$ is always a random variable given that $X$ is a random variable?

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    $\begingroup$ What is your definition of a random variable? Also, what exactly do you not understand here? Why is this question of interest to you? $\endgroup$
    – CrabMan
    Commented Nov 13, 2020 at 15:59

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Within the limited context of what you have provided, and from what I am inferring that you are asking, yes you are correct.

There is nothing random about the absolute value function, it is completely deterministic. Taking a variable $u$ and applying the absolute value function $|\cdot|$ to get the output $w$, that is $w = |u|$, the output $w$ is non-random.

However, the output of a function of a random variable $X$ is random, but that comes from the "randomness" of the random variable, not the function itself (which is completely deterministic). So if we take a random variable $X$, apply the absolute value function $|\cdot|$ to get the output $Y$, that is $Y = |X|$, the output $Y$ is random. Why? Because it is a function of a random variable.

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A random variable is a measurable quantity of our choice. For example, we can say X is a random variable that represents outcome of a dice roll. Similarly we can say Y is a random variable representing if the outcome of a dice roll is even. Its all about what you want to measure. As long as X is measurable Y=|X| is also a random variable measuring some other quantity (in this case absolute value of what X measures).

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Roughly saying, if you consider a variable $X$ such that $$\Pr\{X=x\}=\begin{cases}1&,\quad x=y\\0&,\quad x\ne y\end{cases}$$ then yes, the statement is true, unless counterexamples such as $$ \Pr\{X=1\}=\Pr\{X=-1\}=0.5 $$ exist.

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