I have 2 questions related to this question.
- Why do we do the transformation for $i$ to $j$ in the form of $i = j+1$ in the 4th line of the equation? It seems to me here like it is either done so that we can get the equation back to original Poisson form with a different variable or to correct for the summation interval. I still cannot intuitively understand it.
- In the 6th line, it is written as $\lambda\operatorname E((X+1)^{n-1}).$ But I feel like it should be $\lambda\operatorname E((Y+1)^{n-1})$ because did we not transform the variable in the 4th line of the equation