Question on a formula for Poisson Process

Question

If for a poisson process $P_n(t)= P\left\{X(t)=n\right\}=\frac{(\lambda t)^n}{n!}e^{(-\lambda t)}$,

How do we get the formula for $P\left\{X(t+s)-X(t)=n\right\}$?

Request guide

Answer 1

A Poisson process satisfy the stationary increment property,

Hence

\begin{align}P(X(t+s)-X(t)=n)&=P(X(s)-X(0)=n)\\&=P(X(s)=n) \end{align}