values that can be attained by random variables

Question

Can a discrete random variables takes the values $+ \infty$ and $- \infty$ ? Can someone explain to me this with an example?

Answer 1

This is just a matter of convention; you can define them to be allowed to be $\pm\infty$ or you can define them not to be. The more common definition is to require them to always be finite (that is, they take values in $\mathbb{R}$ rather than $\mathbb{R}\cup\{\infty,-\infty\}$), but either definition can be used. Pretty much all the theory works the same. You have to be careful when talking about things like expectations of random variables that take infinite values (the expectation may be infinite or not defined), but you already have to be careful with that when talking about finite random variables that take an unbounded set of values.

Answer 2

No since $\infty$ is not a number. However, some distributions have possible values $[1, 2, \dots)$. For example, a geometric distribution. Imagine rolling a million sided die until you roll a six. It could take a while.