Final answer:
The mean of W(t) for a Poisson process is λtμ, and its variance involves the calculation of E[W(t)2] and the properties of Xn. The covariance and correlation between W(t) and W(s) use the overlapping intervals and normalization by their variances.
Step-by-step explanation:
The student asks to find the mean and variance of the total effect W(t) for a Poisson process with constant intensity λ and effects Xn. Also, they are requesting the calculation of the covariance and correlation of W(t) and W(s).
Firstly, the mean of the total effect W(t) is calculated using the linearity of expectation and the properties of the Poisson process. Since the effects Xn are assumed to have a mean μ and variance σ2, the mean of W(t) is λtμ.
The variance of W(t) involves more steps as it requires calculating the expected value of the square of W(t) and then using the formula Var(W(t)) = E[W(t)2] - (E[W(t)])2. The calculation would utilize the first two moments of Xn and properties of the Poisson process.
To determine the covariance of W(t) and W(s), we would consider the overlapping intervals of [0, s] and [0, t], assuming without loss of generality that s ≤ t. Using the properties of the Poisson process and the effect Xn, we could compute it in terms of the parameters λ, μ, and σ2.
The correlation between W(t) and W(s) is obtained by normalizing the covariance by the square root of the product of the variances of W(t) and W(s). This is typically expressed using the formula for correlation ρ(W(t), W(s)) = Cov(W(t), W(s))/(Var(W(t))Var(W(s)))1/2.