[Math] Derive the Cramer-Rao lower bound (CRLB) for the variance of any unbiased for estimator of $\lambda$ for random sample from Poisson ($\lambda$)

Question

Let $Y_1,…,Y_n$ be a random sample from Poisson ($\lambda$).

Derive the Cramer-Rao lower bound (CRLB) for the variance of any unbiased for estimator of $\lambda$.

I first set

$$L(\lambda) = \prod_{i=1}^n \frac{\lambda^y e^{-\lambda}}{y!} = \frac{\lambda^{\prod_{i = 1}^n y_i} e^{-\lambda n}}{\prod_{i=1}^n y_i !} .$$

Then setting it to $\ln$ form:

$$\ln L(\lambda) = \frac{\sum_{i=1}^n y_i \ln(\lambda) – \lambda n}{\sum_{i=1}^{n} y_i !}.$$

Can anyone please confirm what I am doing is correct before I move forward?

Answer 1

The likelihood function is $$ L(\lambda)=\left(\prod_{i=1}^n \frac{1}{y_i!}\right)\lambda^{\sum_{i=1}^n y_i}e^{-\lambda n} $$ and hence the log-likelihood function becomes $$ l(\lambda)=-\lambda n+\log(\lambda)\sum_{i=1}^n y_i-\sum_{i=1}^n \log(y_i!). $$