Expected Number of points in Point Poisson Process

Question

Let $\lambda$ be the intensity of points, distributed as point poisson process, in a square grid of area $A$. Then, the Cumulative disributive function is given by: $$ P(r \leq R) = 1 - e^{-\lambda \pi R^2} $$ Using the above cumulative distributive function, how can i calculate the expected number of points, how can i calculate the expected number of points within a radius $R$.

Answer 1

The number of points of a Poisson process with intensity $\lambda$ which are in a domain $D$ of volume $V$ is a Poisson random variable with parameter $\lambda V$. Hence its expectation is $_____$.

(As explained in every decent introduction to Poisson processes.)

Nota: I am unable to ascribe a meaning to this passage from the question:

Then, the Cumulative disributive function is given by: $ P(r \leq R) = 1 - e^{-\lambda \pi R^2} $ Using the above cumulative distributive function...