# Is $|X|$ always a random variable given that $X$ is one? [closed]

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

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

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.
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).
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.