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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. Relevant 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.

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# Bivariate poisson - probability

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