I am working with the following question, which asks for the type of distribution.
Let $X$ be a random variable of distribution Exp($\lambda$). Let $\{x\},\ \lfloor x\rfloor$ denote the fractional part and the integer part of $x\in\mathbb{R}$, which type of distribution does $\{x\}$ have conditional on $\lfloor x\rfloor$?
I have calculated the cumulative distribution function $$F(\theta|k)=\mathbb{P}\left(\{x\}<\theta\ |\ \lfloor x\rfloor = k\right) = \frac{1-e^{-\lambda\theta}}{1-e^{-\lambda}}$$ However, I have searched for many types of distributions and none of them matched this one. I would be really grateful if someone could help. Thanks!