Poisson Process Thinning and Superposition

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on probability modeling, specifically regarding Poisson Processes.

Your Task
Orders arrive at an exchange according to a homogeneous Poisson process $N(t)$ with rate $\lambda > 0$. Each order is independently classified as a Buy order with probability $p \in (0, 1)$ or a Sell order with probability $1-p$.
(1) Prove that the Buy arrival process $N_B(t)$ and Sell arrival process $N_S(t)$ are two independent Poisson processes with rates $\lambda p$ and $\lambda(1-p)$.
(2) Given that exactly $k$ Buy orders arrived in $[0, t]$, find the conditional distribution of total orders $N(t)$ in $[0, t]$.

Constraints

• All variables are defined on a standard probability space $(\Omega, \mathcal{F}, \mathbb{P})$.
• Analytical expressions must be in closed form, and empirical simulations should converge within standard statistical tolerances.