Poisson process probabilities and o notation.

Question

I am struggling with understanding a certain result. I study poisson processes, and a solution to an exercise I am doing does not make sense to me.

Let $N_1(t)$ be a Poisson($\lambda$) process and $N_2(t)$ a Poisson(3$\lambda$) process that are independent. Then

$$P(N_1(h)=1,N_2(h)=0)=(\lambda h+o(h))(1-3\lambda h+o(h))=\lambda h+o(h)$$

What I don't understand is the last equality. How to we derive that? Thank you.

Answer 1

Expand: when $h\to 0$, $$\begin{align} (\lambda h+o(h))(1-3\lambda h+o(h)) &= \lambda h+o(h)-3\lambda^2h^2+o(h^2)+o(h^2) \\ &= \lambda h+o(h) \end{align}$$ since $h^2=o(h)$.