If I have a question such as:

$\displaystyle P(X(1) = 2,X(3) = 12)$ with Poisson Distribution $\displaystyle X - Po(4)$

Would I be right in saying this is the same as:

$\displaystyle P(X(1) = 2,X(3) - X(1) = (12-3))$

=

$\displaystyle P(X(1) = 2,X(2) = 9)$?

Thank in advance

Craig