Just had a bash at this question for my Intro to Maths Stats module...I got to the end with a probability density function rather than a probability mass function, namely $f_Y(y) = \lambda a e^{-\lambda a y}.$ Obviously I'm missing some subtleties with the floor function that makes the new r.v. into a probability mass function instead. Anyways, here it is...
Suppose $X$ is an $\text{exponential}(\lambda)$ r.v. given by 0 for $x$ < 0 and $\lambda e^{-\lambda x}$ for $x \geq 0$. Recall the function $\lfloor x\rfloor$ is defined as the largest integer $n \leq x.$
Let $Y$ be defined by $ Y = \lfloor \frac{X}{a} \rfloor$, where $a$ > 0. Find the probability mass function of $Y$ and hence deduce that $Y$ is a geometric r.v., stating its parameter.
Thanks in advance (should be a quick one!) Sam