Setting:
Defn: for every $x \in \mathbb{R}$ define its negative part by $x^{-} = -x$ if $x \leq 0$, and $x^{-} = 0$ if $x > 0$
Let $\{X_j, j \ge 1\}$, $X_j \overset{d}{\sim} Poisson(1) = \Pr\{X = k\} = \frac{e^{-1}}{k!}$ for $k \geq 0$.
Let $\{S_n , n \ge 1\}$ corresponding sequence of partial sums.
Let $T_n = \frac{S_n - n}{\sqrt{n}}$
I want to show $\pmb{E}[T_n^-] = \frac{n^{n+\frac{1}{2}}e^{-n}}{n!}$
My Solution
First I massaged the expression into a statement about the tails, although I am not sure the result is correct: $$ \pmb{E}[T_n^-] = \pmb{E}[T_n \pmb{1}_{(-\infty,0]} (T_n)] = \pmb{E}\left[ \frac{S_n - n}{\sqrt{n}} \pmb{1}_{(-\infty,0]} (T_n) \right] = \pmb{Pr}\Big\{ \frac{S_n}{\sqrt{n}} - \frac{n}{\sqrt{n}} < 0 \Big\}$$
where I recognized that
$$\pmb{E}\left[ \frac{S_n}{\sqrt{n}} \right] = \frac{1}{\sqrt{n}}\pmb{E}[X_1 + \ldots + X_n] = \frac{1}{\sqrt{n}}(\pmb{E}[X_1] + \ldots + \pmb{E}[X_n]) = \frac{n}{\sqrt{n}}$$
So we have $$\pmb{E}[T_n^-] = \pmb{Pr}\Big\{ \frac{S_n}{\sqrt{n}} - \pmb{E}\left[\frac{S_n}{\sqrt{n}}\right] < 0 \Big\} = 1 - \pmb{Pr}\Big\{ \frac{S_n}{\sqrt{n}} - \pmb{E}\left[\frac{S_n}{\sqrt{n}}\right] \ge 0 \Big\} $$
Now this is where I am really uncertain, what is the variance? my argument is that
$$ Var\left(\frac{S_n}{\sqrt{n}}\right) = \pmb{E}\left[\left(\frac{S_n}{\sqrt{n}} - \pmb{E}\left[\frac{S_n}{\sqrt{n}}\right]\right)^2\right] = \pmb{E}\left[\left(\frac{S_n}{\sqrt{n}} - \frac{n}{\sqrt{n}}\right)^2\right] = \pmb{E}\left[\frac{S_n^2}{n} - 2 \frac{S_n n}{n} + \frac{n^2}{n}\right]\\ = \frac{1}{n} \pmb{E}[S_n^2] - 2 \pmb{E}[S_n] + n = \frac{n^2}{n} - 2 n + n = 0$$
But clearly this cannot be correct since if it is, then we have by Chebyshev's inequality on the last term:
$$\pmb{Pr}\Big\{ \frac{S_n}{\sqrt{n}} - \pmb{E}\left[\frac{S_n}{\sqrt{n}}\right] \ge \epsilon \Big\} \le \frac{Var(\frac{S_n}{\sqrt{n}})}{\epsilon^2} \rightarrow^p \frac{0}{\epsilon^2} = 0$$
Which is not the result I would like to arrive at.
$$\pmb{E}[T_n^-] = 1 - \pmb{Pr}\Big\{ \frac{S_n}{\sqrt{n}} - \pmb{E}\left[\frac{S_n}{\sqrt{n}}\right] \ge 0 \Big\} = 1 - 0 = 1$$
So where did I go wrong? P.S. Stirling's formula may be useful here, but currently I found no use for it.
Hint: Let $S$ denote any Poisson random variable with parameter $n$, then $$\mathrm e^{n}E((n-S)^+)=\sum_{k=0}^n(n-k)\frac{n^k}{k!}=\sum_{k=0}^n\frac{n^{k+1}}{k!}-\sum_{k=1}^n\frac{n^k}{(k-1)!}.$$ The change of variable $i=k-1$ in the last sum on the RHS yields $$\mathrm e^{n}E((n-S)^+)=\sum_{k=0}^n\frac{n^{k+1}}{k!}-\sum_{i=0}^{n-1}\frac{n^{i+1}}{i!}=\frac{n^{n+1}}{n!}.$$ Surely you can conclude from here.