Bivariate poisson - probability

[Milky]

Homework Statement

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))

Homework Equations

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.

 

[mighty2000]

Is my reasoning correct?

Your approach to part (a) is correct. By independence, you can rewrite the joint probability as the product of the individual probabilities. However, you may want to consider conditioning on the value of M2(t) in order to simplify the calculation. This is because M2(t) appears in both N1(t) and N2(t), and conditioning on it will remove it from the probabilities.

For part (b), your approach is also correct. You can use the formula for covariance and then substitute in the expected values of N1(t) and N2(t) that you have calculated. Your reasoning is also correct, as long as you assume that M1(t), M2(t), and M3(t) are independent. Your final expression for the covariance looks correct to me.

Keep up the good work!