probability - what is $|X|$?

Question

In probability ,

Let $X$ be an independent random variable $X$.

When someone writes $|X|$-what does he mean?

Thank you.

Answer 1

It doesn't make sense to say $X$ is independent. You can say $X$ is independent of another random variable $Y$, or more rarely of some event, but not that it is simply independent.

A random variable is a (measurable) function. $|X|$ means you compose the functions, so you apply the absolute value function $|\cdot|$ to the value of $X$.

For example, suppose you roll a fair die, and $X$ is the value shown on the die minus $10$. That means $X$ takes the values $-9$, $-8$, $-7$, $-6$, $-5$, and $-4$ each with probability $1/6$.

• The random variable $X+1$ takes the values $-8$, $-7$, $-6$, $-5$, $-4$, and $-3$ each with probability $1/6$.

• The random variable $X^2$ takes the values $81$, $64$, $49$, $36$, $25$, and $16$ each with probability $1/6$.

• The random variable $|X|$ takes the values $9$, $8$, $7$, $6$, $5$, and $4$ each with probability $1/6$.