Suppose we have a random variable $X$ and we know its distribution. How would we determine the distribution of $Y:=K\times\lfloor \frac{X}{K}\rfloor$ in relation to $X$, where $K$ is just any constant?
One way I can think of is through its cdf $\mathbb{P}(Y<y)=\mathbb{P}({\lfloor \frac{X}{K}\rfloor}<\frac{y}{K}).$ But how do I simplify this expression further?
I think this probability is $\mathbb{P}(K\lfloor \frac{y}{K}\rfloor<X<K\lfloor \frac{y}{K}\rfloor+K),$ then we can express this via the cdf of $X$. However I am not sure if this is the correct approach?
Many thanks in advance!
Rather than approaching the problem through the cdf of $Y$, I'd suggest directly finding the pdf. $\mathbb{P}(Y=y)$ is $0$ if $y$ isn't an integer multiple of $K$. Otherwise, $$\mathbb{P}(Y=nK) = \mathbb{P}\left(K \left\lfloor \frac{X}{K} \right\rfloor = nK \right) = \mathbb{P}\left(nK \le X < (n+1)K\right)$$
There won't be a simple closed form for this without knowing the specific distribution of $X$.