Poisson process

Let W(t) be a Brownian motion and let Q(t) be a compound Poisson process, both defined on the same probability space (Ω,ℱ, ℙ) and relative to the same filtration ℱ (t) , t ≥0. Show that, for each t, the random variables W(t) and Q(t) are independent. (In fact, the whole path of W is independent of the whole path of Q, although you are not being asked to prove this stronger statement.)