Distribution of $X(t+s)$ in Poisson proccess in terms of distribution of $X(t)$ and $X(s)$

Question

Let $(X(t))_{t \geq0}$ be a Poisson process with intensity $\lambda$. Denote $p_t$ as distribution of $X(t)$. Write $p_{t+s}$ in terms of $p_t$ and $p_s$.

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This was a homework we skipped due to insufficient time and the lecturer kinda forgot about it next time, so it is left unsolved. I appreciate any help as I wasn't really able to work it out.

Answer 1

$X(t+s)=(X(t+s)-X(s))+X(s)$. $X(t+s)-X(s)$ and $X(s)$ are independent and their distributions are $\rho_s$ and $\rho_t$. Hence the distribution of $X(t+s)$ is the convolution of $\rho_t$ and $\rho_s$.