QuestionQuestion: Poisson Distribution
Let $X_{1}, X_{2}, \ldots, X_{n}$ be independent and identically distributed (i.i.d.) random variables, where each $X_{i}$ follows a Poisson distribution with parameter $\lambda>0$. The probability mass function (PMF) for a Poisson random variable is given by:
Likelihood Estimation fo $$f_{X}(x ; \lambda)=\frac{\lambda^{x} e^{-\lambda}}{x!}, \quad x=0,1,2, \ldots$$ where $\lambda$ is the rate parameter of the Poisson distribution. (a) Write the likelihood function $L(\lambda)$ for the sample $X_{1}, X_{2}, \ldots, X_{n}$. (b) Derive the log-likelihood function $\ell(\lambda)=\ln L(\lambda)$. (c) Find the Maximum Likelihood Estimator (MLE) for $\lambda$ by solving $\frac{\partial e(\lambda)}{\partial \lambda}=$ 0 . (d) Verify that the second derivative of the log-likelihood function at the MLE is negative, confirming that the MLE is indeed a maximum. (e) Find the Fisher information for $\lambda, I(\lambda)=-E\left[\frac{\partial^{2} \ell(\lambda)}{\partial \lambda^{2}}\right]$. (f) Using the MLE and Fisher information, calculate the Cramer-Rao lower bound for the variance of the MLE.