Find conditional probability of given Poisson process

Question

A Poisson process have rates $\lambda$, and given $0<s<t$.

I want to find $P(X(t)=n+k\vert X(s)=n)$.

I have tried like this \begin{eqnarray} P(X(t)=n+k\vert X(s)=n) &=& \dfrac{P(X(t)=n+k,X(s)=n)}{P(X(s)=n)}\\ &=& \dfrac{P(X(t)-X(s)=n+k-n=k)}{P(X(s)=n)}\\ &=& \dfrac{P(X(t-s)=k)}{P(X(s)=n)}\\ &=& \dfrac{\dfrac{e^{-\lambda(t-s)}(\lambda(t-s))^k}{k!}}{\dfrac{e^{-\lambda s}(\lambda s)^n}{n!}}. \end{eqnarray}

It is correct or incorrect?

Answer 1

$\frac {P(X(t)=n+k,X(s)=n)} {P(X(s)=n)}= \frac {P(X(t)-X(s)=k,X(s)=n)} {P(X(s)=n)}=P(X(t)-X(s)=k)$ (by independence of $X(s)$ and $X(t)-X(s)$) $=e^{-\lambda (t-s)} \frac {\lambda (t-s)^{k}} {k!}$.