Describe the distribution measure of a random variable

Question

Considering a discrete probability space $\mathbb{N} = \{0,1,2,...\}$ such that

$$P(\{n\}) = (1-q) \ q^n$$

where $P$ denotes the probability measure and $q$ is a fixed number from the interval $(0,1)$ and $\epsilon: \mathbb{N} \to \mathbb{R}$ is a random variable such that

$$\epsilon(n) = (-1)^n$$

How can I describe the distribution measure of $\epsilon$ on $\mathbb{R}$?

I am not sure I understand it correctly, but I assume that the possible outcomes are $\mathbb{N} = \{0,1,2,...\}$ and for every possible outcome there is a probability $P(\{n\}) = (1-q) \ q^n$ for it to occur.

But I don't know what it means that $\epsilon(n) = (-1)^n$ and how to sufficiently describe the distribution measure of $\epsilon$.

Answer 1

$\epsilon$ takes only the values -1 and +1. it takes the value +1 on the set $\{0,2,4,...\}$ and -1 on $\{1,3,5,...\}$. $P\{\epsilon =1\}=\sum_n P(\{2n\})=\sum_n (1-q)q^{2n}=\frac {1-q} {1-q^{2}}=\frac 1 {1+q}$ and $\epsilon = -1\}= \sum_n (1-q)q^{2n-1}=\frac q {1+q}$.