Poisson probability

Question

Given that $N(t)$ follows a Poisson process with rate $\lambda t$, How can I compute the following probability $P(2N(7) - N(1)=3)$?

It seems that I can't use the property that the sum of independent Poisson variables is Poisson with the sum of the rates, because $N(7)$ and $N(1)$ don't seem independent. Any help is appreciated!

Answer 1

Hints:

• $N(1)$ and $N(7)-N(1)$ are independent and $N(7)-N(1)$ has Poisson distribution with parameter $6\lambda$.
• $2N(7)-N(1)=2[N(7)-N(1)]+N(1)$