# How do I transform an r.v. using the floor function? (exponential distribution)

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

Hint:* $Y$ is clearly discrete taking values in the set of non-negative integers, due to the the flooring. Then, for any integer $n\geq 0$ we have $$P(Y = n) = P\left(X\in [an, a(n+1))\right)$$ $$= \int_{an}^{a(n+1)} \lambda\mathrm e^{-\lambda x}dx= (1-p)^np$$ where $p = 1 - e^{-\lambda a} \in (0,1),$ as $\lambda>0$ and $a>0$.
$Y$ takes on value $k \in \{0,1,\ldots,\}$ if $k = \lfloor \frac{X}{a} \rfloor$. Thus, $P(Y=k) = P( \lfloor \frac{X}{a} \rfloor = k) = P(k \leq \frac{X}{a} < k+1) = P(ak \leq X < a(k+1)) = \int_{ak}^{a(k+1)} f(x) dx$ where $f(x)$ is the density of $X$.