I need to find mathematical expectation of the number of points of Poisson process with parameter $\lambda > 0$ such that: these points $\in [1,2]$ and there are no points of Poisson process in their left $\delta$ neighborhood, where $\delta \in (0,1)$. What i mean is: find mathematical expectancy of the number of points such that: for example: if a is the point, then there are not points of Poisson process in $(a - \delta, a)$ and a should be in $[1,2]$
UPD: am i right that the answer is 0, because $N_{t} \in Z$, hence it cannot have values in (0,1) or (1,2)?