What is the Fisher information of a sequence of independent Poisson random variables?

Question

$X_1,...,X_n$ are independent Poisson random variables with$ X_j $having parameter$j\lambda$.What is the fisher information contained in $(X_1,...,X_n)$ about $\lambda$? BTW,What is the likelihood function in this question? $S_n$ is the log likelihood function,is the first derivative right ?i dont know if it is correctas this is the first step to get the answer. What i calculate is $\frac{\partial S_n}{\partial x}=\sum_{i=1}^{n}\frac{X_i}{\lambda i}-1$

Answer 1

Try the following:

1) Calculate the likelihood function based on observations $x_1,\ldots,x_n$ from $X_1,\ldots,X_n$. This is just $$ L(\lambda)=L(\lambda;(x_1,\ldots,x_n))=\prod_{i=1}^n p_i(x_i), $$ where $p_i$ denotes the probability function corresponding to $X_i$. Then calculate the loglikehood function $l(\lambda)=l(\lambda;(x_1,\ldots,x_n))=\log(L(\lambda;(x_1,\ldots,x_n)))$.

2) Differentiate twice with respect to $\lambda$ and get an expression for $$ \frac{\partial^2 l(\lambda)}{\partial \lambda^2}. $$

3) Then the Fischer information is the following $$ i(\lambda)=E\left[-\frac{\partial^2 l(\lambda;(X_1,\ldots,X_n)}{\partial \lambda^2}\right]. $$

I think the correct answer must be $\frac{n(n+1)}{2}\frac{1}{\lambda}$, but please correct me if I'm wrong.