Describe distribution measure probability So we will consider the probability space (discrete case) where $\mathbb{N}=\{0,1,2,...\}$. We also have $T(\{n\})=(1-x)x^n$, where $T$ denotes the probability measure and where $x$ is a fixed number for $x\in (0,1)$. Now let $\lambda:\mathbb{N}\rightarrow \mathbb{R}$ be a R.V. such that
\begin{equation}
\lambda(n)=\text{the remainder from division of }n \text{ by } 3
\end{equation}
a) Describe the distribution measure of $\lambda$ on $\mathbb{R}$.

I know the remainders will be 0,1 and 2 and it continues like that, but I don't see how to proceed further. 

To not leave the question unanswered. As by the comment by Mike Earnest:
$$\mathbb{P}_\lambda(3\mathbb{N})=\mathbb{P}(\lambda \in 3\mathbb N)=\mathbb{P}(\cup_{k\in \mathbb{N}}\lambda =3k)=\sum_{k\in \mathbb{N}}\mathbb{P}(\lambda=3k)=\sum_k (1-x)x^{3k}$$
Similarly  for $\mathbb{3\mathbb N+1}$ and $3\mathbb{N}+2$.
