I am currently working with a rounded up values of continuous distribution and I am trying to find out some of its general properties. I have the following model:
$y = \lceil x \rceil$
where $x$ has continuous distribution and $\lceil x \rceil$ is a rounded up value of $x$. I am mainly interested in quantiles of distribution of $\lceil x \rceil$. While the quantiles of $x$ are straight forward, the rounded up version seems trickier. I could do simulations, but I have a feeling that there is a much simpler and neater way of working with its quantiles. So if we define quantiles $n$ and $k$ the following way:
$k: P(x < k) = 1 - \alpha,$
$n: P(\lceil x \rceil < n) \geq 1 - \alpha,$
then I have a very strong feeling that the following is true:
$\lceil k \rceil = n$
This means that the quantiles of rounded up $x$ are equal to rounded up quantiles of $x$. The simulations I've done in R so far support this finding and I cannot find any case, where this would be violated. However I need an appropriate mathematical proof for this, and I can't seem to find one. And, by the way, in case this is not always true, I need to know the conditions when this equation does not hold.
Does anyone have any references to the proof or any ideas of how to prove this?