Hello,

No you don't need.

Y has integer values.

So what's the probability for Y to be equal to an integer k ?

P(Y=k)=P([x]+1=k)=P([x]=k-1)

For [x] to be equal to k-1, that means that x is between k-1 and k (definition).

So P(Y=k)=P(k-1<x < k)

There's no way of simplifying it, but this defines Y's distribution.