Renaming random variables willy-nilly I am looking at the following: < I get everything until the last step, right after the line with (added term is zero). I understand that if you have X ~ Poisson($\lambda$) it doesn't matter what you call the actual variable in the mass function, $$\frac{\lambda^x e^{- \lambda}}{x!} = \frac{\lambda^z e^{- \lambda}}{z!} = \frac{\lambda^k e^{- \lambda}}{k!}$$ but in this case, y was already defined as y = x + 1, and suddenly it was redefined as x a few lines later? Why is this allowable? Where else can I do such things? Thanks!

In the last line, $X$ is the random variable. In the next-to-last line, $y$ is a dummy variable that steps through all the possible values of $X$, so it can be called anything. It could even be called $x$, but the author changed the name of the dummy to explain the substitution more clearly to the reader.

The important thing about $x$ and $y$ is that the summand takes on exactly the same set of values as $x$ and $y$ step through their specified values: as $x$ steps through its range $0,1,2,\ldots$, the value of $x+1$ in the summand takes the values $1,2,3,\ldots$; as $y$ steps through its range $1,2,3,\ldots$, the value of $y$ in the summand (which appears where $x+1$ appeared before) takes on the same set of values $1,2,3,\ldots$. So in the end, the same set of summands occurs in both cases.