Cant understand skipped step in textbook solution

Question

I can't understand this solution from my textbook. somehow $e^{tn}$ comes out of this expectation. But I don't understand why because inside the expectation it is $E(e^{(s-t)X_c}e^{tx}|x=n)$. I thought that $E(e^{tx})$ should produce the Poisson moment function, but $e^{tn}$ is not the Poisson moment function... So I have no clue what the author did there. Can anyone explain this step?

Answer 1

The expectation operates only on random variables. Also in general:

$$\mathbb{E}[aX+b]=a\mathbb{E}[X]+b$$

This is due to the fact that $a$ and $b$ are not random variables. In the case given by the book, $e^{tx}$ is the $a$ in the above expression. $e^{tx}$ is indeed NOT a random variable, since $x$ is a constant (it is given that $x=n$), so it can just be taken out of the expectation. So in other words, $e^{tx}=e^{tn}$...which is just a number

Answer 2

$E[e^{s X_c + t(X-X_C)}|X=n]=E[e^{s X_c + t n-t X_C)}|X=n]$

So $e^{tn}$ is just a scalar, which can be taken outside $E$ since expectations are linear.

Answer 3

Once you condition on $\{X=n\}$, $e^{tX}$ becomes the constant $e^{tn}$ and it can be taken out of the expectation.