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

Question

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

Answer 1

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 $a>0$.

Answer 2

If $k = \left\lfloor \frac{X}{a} \right\rfloor$, $Y$ takes on non negative integer values $$k \in \{0,1,\ldots,\}$$

Thus, the probability that we want is: $$P(Y=k) = P\left( \left\lfloor \frac{X}{a} \right\rfloor = k \right) $$

Removing the floor function gives: $$= P\left(k \leq \frac{X}{a} < k+1\right) = P\left(ak \leq X < a(k+1)\right)$$

And converting to an integral gives: $$= \int_{ak}^{a(k+1)} f(x) dx$$ where $f(x)$ is the density of $X$.