Determine a system items arrive to a poisson process

Question

Assignment:

Consider a system where items arrive according to a Poisson process with parameter λ = 2 item/sec. Let X(t) denote the number of items arrived in the time interval [0, t) for any t ≥ 0. Assuming that X(0) = 0, determine the following probabilities:

a) P [X(1) = 2];

b) P [X(1) = 2, X(3) = 6];

c) P [X(1) = 2|X(3) = 6];

d) P [X(3) = 6|X(1) = 2].