Does it make sense to round a continuous random variable to an integer value?

Question

I need to get discrete random variables, I know about the existence of distributions of discrete RVs, but I want to get something similar to the exponential distribution. Does it make sense to get an exponentially distributed RV and round it up to an integer value? Can I say that such discrete values are obtained according to the exponential law, or is this erroneous? On a sample of several thousand, I get a histogram very similar to an exponential distribution law

Answer 1

If $X$ has an exponential distribution with mean $\lambda$, $\lceil X \rceil$ has probability mass function

$$\mathbb P(\lceil X \rceil = k) = \mathbb P(k-1< X \le k) = \left(e^{1/\lambda}-1\right) e^{-k/\lambda}$$ for positive integers $k$, which makes it a geometric random variable with parameter $p =1 - e^{-1/\lambda}$.