To show a random variable is geometric

Question

If $X$ is exponential with rate $\lambda$, show that $Y= [X]+1$ is geometric with parameter $p= 1 - e^{-\lambda}$, where $[x]$ is the largest integer less than or equal to $x$.

I dont know hot to approach. I understand that $[X]$ will give a step function with integer values ...

Answer 1

Notice that $$\{[X] = x\}\iff\{x\leq X<x+1\}.$$ Then you want to examine the $P(Y = y)$, which is

$$P(Y=y) = P([X]+1 = y) = P([X] = y-1).$$

Hopefully, you can take it from there.