Find the Fisher Information of $X\sim{\rm Poisson}(\mu)$

Question

Suppose $X \sim Poisson(\mu)$, so for $i \in \{0,1,2,\ldots\}$, $P(X = i) = \exp(-\mu) \mu^i/i!.$

Find the Fisher Information of $X$.

This is what I have done so far

$S_X(i) = \frac{\partial \ln (f_{X|\mu}(i))/\partial \mu}{f_{X|\mu}(i)} = -1 + \frac{i}{\mu}$

How do I find the probabilities to get the fisher score?

Answer 1

$f(x|\mu) = \exp(-\mu) \mu^x/x!$

$\implies l(x|\mu) = \ln{f(x|\mu)} = -\mu + x\ln{\mu} - \sum_{i=1}^{x}i$

$l^{\prime}(x|\mu) = -1 + \frac{x}{\mu}$

$l^{\prime\prime}(x|\mu) = - \frac{x}{\mu^2}$

Fisher information $I(\mu) = -E[l^{\prime\prime}(x|\mu)] = \frac{E[X]}{\mu^2} = \frac{1}{\mu} $