Move Score and Move Information of the Poisson Probability Mass Function

Let $\mathbb{N}$ be the set of natural numbers, $\mathbb{N}_0=\mathbb{N}\cup\{0\}$, and let $\mathbb{R}$ be the set of real numbers, identified with the Euclidean space $\mathbb{R}^{1}$. We use the factorial $k!$ with the conventions $0!=1$ and $\mu^{0}=1$, and the exponential function $\exp$.

Fix a real number $\mu>0$ and define $p_\mu:\mathbb{R}\to\mathbb{R}$ by

$$p_\mu(k)=\exp(-\mu)\frac{\mu^{k}}{k!}\quad(k\in\mathbb{N}_0),\qquad p_\mu(x)=0\quad(x\in\mathbb{R}\setminus\mathbb{N}_0),$$

and let $P_\mu$ be the Poisson distribution with parameter $\mu$. Let $w_1\in\mathbb{R}$, and consider the single move $a=(a_1)$ with $a_1=1$ and the single weight $w=(w_1)$, in the sense of Move Score of a Discrete Probability Mass Function.

1. $p_\mu(k)=P_\mu(\{k\})$ for every $k\in\mathbb{N}_0$, and $p_\mu$ is a discrete probability mass function on $\mathbb{R}$ with $\{x\in\mathbb{R}:p_\mu(x)>0\}=\mathbb{N}_0$.

2. The move score of $p_\mu$ for $a$ and $w$ is

$$\rho_{p_\mu,a,w}(k)=w_1\Bigl(1-\frac{k}{\mu}\Bigr)\qquad(k\in\mathbb{N}_0),$$

and $\{k\in\mathbb{N}_0:\ k+a_1\notin\mathbb{N}_0\}=\emptyset$.

3. The move information of $p_\mu$ for $a$ and $w$ is

$$\mathsf{J}(p_\mu;a,w)=\frac{w_1^{2}}{\mu}.$$