I have a continuous random variable $X$ (positive). I want to simulate its distribution with a discrete distribution and calculate $E[X]$ from that discrete distribution. So, the obvious approach is to divide the range of the random variable into step size of $h$. Let the CDF values at the points $0, h, 2h, \ldots$ be $P_0, P_1, P_2, \ldots$.
So, $\mathrm{Prob}(0 < X \leq h) =P_1-P_0$, $\mathrm{Prob}(h < X \leq 2h) = P_2-P_1$; and so on.
Now these probability masses are associated with a interval. We need to find a representative point of each interval. For an interval $(a,b]$ which point should we take as the representative point leftmost point, rightmost point or the mid point to get a good estimate of $E[X]$. This is my question.