Expected value of a truncated binomial Let $X\sim B(n,p)$ be a binomial random variable and fix $0<k<n$.  Are there any well-known bounds for $\mathbb{E} (X-k)^+$, where $(X-k)^+ =\max\{0,X-k\}$?  I am particularly interested in interpreting this as a function of $p\in (0,1)$.
 A: $\newcommand\vpi{\varphi}$Let 
$$Z_n:=\frac{X_n-np}{\sqrt{npq}},$$
where $X_n:=X$ and $q:=1-p\in(0,1)$. Then $Z_n\to Z\sim N(0,1)$ in distribution (as $n\to\infty$). Also, $EZ_n^2=1$. So,
assuming that 
$$z_{n,k}:=\frac{k-np}{\sqrt{npq}}\to z\in\mathbb R, \tag{0}$$
by the uniform integrability we have 
$$\frac{E(X-k)^+}{\sqrt{npq}}=E(Z_n-z_{n,k})^+
\to E(Z-z)^+,$$
whence 
$$E(X-k)^+\sim\sqrt{npq}\, E(Z-z)^+. \tag{1}$$

Also, note that for real $z$
$$\psi(z):=E(Z-z)^+=\int_z^\infty(u-z)\vpi(u)\,du=\vpi(z)-z(1-\Phi(z)),$$
where $\vpi$ and $\Phi$ denote, respectively, the pdf and cdf of $N(0,1)$. Here is the graph $\{(z,\psi(z))\colon|z|<3\}$: 
 

Using more advanced tools than the central limit theorem for the binomial distribution, one can greatly improve estimate (1) (derived assuming (0)). Indeed, by the nonuniform Berry--Esseen bound (see e.g. formula (4.3.1)), for all real $t$
$$P(Z_n>t)-P(Z>t)=O\Big(\frac1{(1+|t|^3)\sqrt{npq}}\Big). \tag{2}$$
Everywhere here the constants in $O(\cdot)$ are universal. Also, for any real-valued random variable $Y$ and any real $y$,
$$E(Y-y)^+=E\int_y^\infty du\, 1_{Y>u}
=\int_y^\infty du\, P(Y>u).$$
So, by (2),
$$E(X-k)^+=\sqrt{npq}\,E(Z_n-z_{n,k})^+
=\sqrt{npq}\,E(Z-z_{n,k})^+ + O(1). \tag{3}$$
Of course, (3) implies (1) (when (0) holds), but (3) is much stronger and more general than (1). 
A: The command of Mathematica 12.0
Mean[TransformedDistribution[Max[x - k, 0],x\[Distributed] BinomialDistribution[n,p]]]

produces $${p^{\lfloor k\rfloor +1} (1-p)^{n-\lfloor k\rfloor } \left(-\binom{n}{\lfloor k\rfloor +1} \, _2F_1\left(1,-n+\lfloor k\rfloor +1;\lfloor k\rfloor +2;\frac{p}{p-1}\right)+k \binom{n}{\lfloor k\rfloor +1} \, _2F_1\left(1,-n+\lfloor k\rfloor +1;\lfloor k\rfloor +2;\frac{p}{p-1}\right)-k p \binom{n}{\lfloor k\rfloor +1} \, _2F_1\left(1,-n+\lfloor k\rfloor +1;\lfloor k\rfloor +2;\frac{p}{p-1}\right)+p \binom{n}{\lfloor k\rfloor +1} \, _2F_1\left(1,-n+\lfloor k\rfloor +1;\lfloor k\rfloor +2;\frac{p}{p-1}\right)+p \lfloor k\rfloor  \binom{n}{\lfloor k\rfloor +1} \, _2F_1\left(1,-n+\lfloor k\rfloor +1;\lfloor k\rfloor +2;\frac{p}{p-1}\right)-\lfloor k\rfloor  \binom{n}{\lfloor k\rfloor +1} \, _2F_1\left(1,-n+\lfloor k\rfloor +1;\lfloor k\rfloor +2;\frac{p}{p-1}\right)-p \binom{n}{\lfloor k\rfloor +2} \, _2F_1\left(2,-n+\lfloor k\rfloor +2;\lfloor k\rfloor +3;\frac{p}{p-1}\right)\right)}{(p-1)^{-2}} $$ for $k\ge 0,\, k <n$.
