Question

Let N1(t), N2(t) and N3 be independent Poisson process with parameters 1 = 1, 2 = 2 and 3 = 3. Consider two process, X(t) = N1(t)N3(t) and Y(t) = N2(t)N3(t) . Prove that the processes X(t) and Y (t) are correlated and find the correlation coecient of X(t) and Y (t).

Answer 1

To prove that the processes X(t) = N1(t)N3(t) and Y(t) = N2(t)N3(t) are correlated, we need to find the correlation coefficient between them.

Step-by-step explanation:

To prove that the processes X(t) = N1(t)N3(t) and Y(t) = N2(t)N3(t) are correlated, we need to find the correlation coefficient between them. The correlation coefficient is given by the formula:

r = Cov(X, Y) / sqrt(Var(X) * Var(Y))

First, we calculate the covariance Cov(X, Y) as:

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

Since N1, N2, and N3 are independent Poisson processes, their expectations are given by:

E(N1(t)) = λ1t = t, E(N2(t)) = λ2t = 2t, and E(N3(t)) = λ3t = 3t

By substituting these values, we get:

E(X) = E(N1(t)N3(t)) = E(N1(t))E(N3(t)) = t * 3t = 3t^2

E(Y) = E(N2(t)N3(t)) = E(N2(t))E(N3(t)) = 2t * 3t = 6t^2

E(XY) = E(N1(t)N3(t)N2(t)N3(t)) = E(N1(t)^2)E(N2(t)^2) = (λ1t + (λ1t)^2)(λ2t + (λ2t)^2) = (t + t^2)(2t + 4t^2)

By substituting these values into the covariance formula, we can calculate the covariance. Then, we can find the variances of X and Y using the fact that the variances of independent random variables are additive. Finally, we substitute the covariance and variances into the correlation coefficient formula to find the correlation coefficient of X(t) and Y(t).