Let $X$ have an exponential distribution with parameter $1$. Let $\lceil x\rceil$ denote the smallest integer greater than $x$, for example $\lceil3.5\rceil=4$ ,$\lceil5\rceil=5$. What is $E\left(\lceil X\rceil\right)$?
I don't quiet understand how to get started on this question. I know the $E[x]$ of a exponential distribution is $\lambda=1/\mu$. After that, I don't know what else I can do.
Let $Y = \lceil X \rceil$. Then what is $\Pr[Y = 1]$? It is simply $$\Pr[Y = 1] = \Pr[\lceil X \rceil = 1] = \Pr[0 < X \le 1] = 1 - e^{-1}.$$ Now ask, what is $\Pr[Y = 2]$? In fact, what is $\Pr[Y = y]$ for some positive integer $y$? It is just $\Pr[y-1 < X \le y]$. Once you get this, how do you calculate the expectation of $Y$?