property of poisson process

Question

I am reading a probability book and I have the following question about the poisson process. Let $N(t), t\ge 0$ be a poisson process with parameter $\lambda$. Then we know that $N(t+s)-N(t)$ does not depend on $t$. Let $s=t$, we get $$N(2t)-N(t)=N(t)-N(0)=N(t). $$ Then we see that the two random variables $N(2t)$ and $2N(t)$ are the same. But obviously, for a general $k$, $$P(N(2t)=k)\ne P(N(t)=k/2). $$ Can somebody point out to me my misunderstanding?

Answer 1

It is not at all correct to say that $N(2t)-N(t)=N(t)-N(0)=N(t).$

Rather, it is the probability distributions of $N(2t)-N(t)$ and $N(t)-N(0)$ that are equal.

Nor is $N(2t)$ the same as $2N(t),$ and in that case even the probability distributions are not equal, although the expected values are. Every value of $2N(t)$ is an even integer, and that is not true of $N(2t).$