Consider a sequence of positive real numbers $(a_n)_{n\in\mathbb{N}}$ and a discrete distrbution $\pi$ on $\mathbb{N}$. Assume that there is a $n_0$ such that for $n<n_0$ it holds $a_n > n$ and for $n>n_0$ then $n>a_n$. Furthermore, and most importantly, we have for some $\pi$ distributed random variable $X$ the following identitiy $$\mathbb{E}[X] =\sum_{n\geq 0} n\pi_n =\sum_{n\geq 1} n\pi_n = \sum_{n\geq 1} a_n \pi_n,$$ where we assume that the expected value exists.
I want to deduce properties of $\pi$ from this identity. I guess the value $n_0$ plays and important role, maybe being the mode of $\pi$ (???). Are there some well known facts for this type of problems? I guess it is also related to analysis and the theory of series... What results are there and/or what might be a good source to use?