Bivariate poisson - probability

[Milky]

Let {Mi(t), t >= 0 }, i=1, 2 be independent Poisson processes with respective rates λi, i=1, 2, and set

N1(t) = M1(t) + M2(t), N2(t) = M2(t) + M3(t)

The stochastic process {(N1(t), N2(t)), t >= 0} is called a bivariate Poisson process.
(a) Find P{N1(t) = n, N2(t) = m}
(b) Find Cov (N1(t), N2(t))



2. Homework Equations



3. The Attempt at a Solution
I am trying to solve this problem as follows:
(a)
P{M1(t) +M2(t) = n, M2(t) +M3(t) = m} = P{M1(t) +M2(t) = n | M2(t) +M3(t) = m} / P{ M2(t) +M3(t) = m}
which then equals by independence of M1(t) +M2(t), M2(t) + M3(t)
= P{M1(t) +M2(t) = n}P{M2(t) +M3(t) = m} / P{ M2(t) +M3(t) = m}

Now, are they independent? Or is my assumption wrong here? I'm starting to think I should condition on the value of M2(t) here, but conditioning on one of 2 variables is starting to confuse me!

AND

(b) By using the fact that:

Cov (X, Y) = E[XY] - E[X]E[Y]

(substituting in for N1(t) and N2(t))

Cov (N1(t), N2(t)) = E[N1(t)N2(t)] - E[N1(t)]E[N2(t)]

and I can get:
E[N1(t)] = E[M1(t) + M2(t)] = E[M1(t)] + E[M2(t)] = (λ1t) + (λ2t)
E[N2(t)] = E[M2(t) + M3(t)] = E[M2(t)] + E[M3(t)] = (λ2t) + (λ3t)

Is my following assumption correct?
E[N1(t)] N2(t)] = E[{M1(t) + M2(t)}{M2(t) + M3(t)}]
= E[M1(t)M2(t) + M2(t)M2(t) + M1(t)M3(t) + M2(t)M3(t)]
= E[M1(t)]E[M2(t)] + E[M2(t)]E[M2(t)] + E[M1(t)]E[M3(t)] + E[M2(t)]E[M3(t)]
= (λ1t)(λ2t) + (λ2t)(λ2t) + (λ1t)(λ3t) + (λ2t)(λ3t)

which would give me the Covariance? Or should I solve this by using the Conditional Covariance formula? I am assumming that M1(t), M2(t), and M3(t) are all independent.

 

[mmwave]

Your approach for part (a) is correct. You can use the fact that the Poisson processes are independent to calculate the probabilities.

For part (b), your approach is also correct. You can use the fact that the Poisson processes are independent to calculate the expected values. However, you should be careful with the calculation of E[N1(t)N2(t)]. It should be:

E[N1(t)N2(t)] = E[(M1(t) + M2(t))(M2(t) + M3(t))]
= E[M1(t)M2(t) + M1(t)M3(t) + M2(t)M2(t) + M2(t)M3(t)]
= E[M1(t)]E[M2(t)] + E[M1(t)]E[M3(t)] + E[M2(t)]E[M2(t)] + E[M2(t)]E[M3(t)]
= (λ1t)(λ2t) + (λ1t)(λ3t) + (λ2t)(λ2t) + (λ2t)(λ3t)

Then, you can substitute these values into the formula for Cov(N1(t), N2(t)).