Given that $X$ is random variable that takes values:
$0,\dots,H-1$
The PMF of $X$ is unknown, but I can tell what is the expected value which is $\bar{X}$
There is event $Y$ when calculated it gives the value:
$P(Y)=E[\frac{1}{X+1}]$
Is there a way to find expected value $\bar{Y}$?
Best Answer
$$\begin{align*} E\left[\frac{1}{X+1}\right] &= 1\cdot p(0) + \frac{1}{2}\cdot p(1) + \frac{1}{3}\cdot p(2) + \cdots + \frac{1}{H}\cdot p(H-1)\\ &\leq p(0) + 1\cdot p(1) + 2\cdot p(2) + \cdots + (H-1)\cdot p(H-1)\\ &= p(0) + E[X]\\ &\leq 1 + E[X]. \end{align*} $$ The weak upper bound $E[\frac{1}{X+1}] \leq p(0) + E[X]$ requires some knowledge of the pmf of $X$, though the even weaker upper bound $1 + E[X]$ requires knowing only the expected value of $X$ which the OP claims he knows, or does he mean that he has the sample mean $\bar{X}$ available to him?