Prove that the process defined below is a martingale Zt := (Nt - λt)2 -λt where Nt is a Poisson Process
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Prove that the process defined below is a martingale
Zt := (Nt - λt)2 -λt
where Nt is a Poisson Process
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- let X be a real-valued compound Poisson process with Lévy measure v, satisfying v( mathbb R )=c< infty . Show that M t :=X t -t int xv(dx),t t >= 0 is a martingale.Prove the following property of the compound Poisson process:1. it has independent and stationary increments.Show that the following series is divergent
- Assume X ∼ Poisson(r), where r > 0. Prove that E(X) = r. Show all the steps of the proof.Let (Nt)t≥0 be a Poisson process. Explain what is wrong with the following proof that N3 is a constant.E ((N3)2) = E(N3N3) = E(N3(N6-N3)) = E(N3)E(N6-N3) = E(N3)E(N3) = E(N3)2.Thus, Var(N3) = E((N3)2)-E(N3)2=0, which gives that N3, is a constant with probability 1.Consider the following configuration of solar photovoltaic arrays consisting of crystalline silicon solar cells. There are two subsystems connected in parallel, each one containing two cells. In order for the system to function, at least one of the two parallel subsystems must work. Within each subsystem, the two cells are connected in series, so a subsystem will work only if all cells in the subsystem work. Consider a particular lifetime value t0, and suppose we want to determine the probability that the system lifetime exceeds t0.Let Ai denote the event that the lifetime of cell i exceeds t0(i = 1, 2, , 4). We assume that the Ai's are independent events (whether any particular cell lasts more than t0 hours has no bearing on whether or not any other cell does) and that P(Ai) = 0.6 for every i since the cells are identical. Using P(Ai) = 0.6, the probability that system lifetime exceeds t0 is easily seen to be 0.5904. To what value would 0.6 have to be changed in order to increase…
- the function f(x)=xln(x-1) has a taylor series expansion at x=1Give a recursive definition (with initial condition(s)) of (an) where (n = 1, 2, 3, . . . )Consider a series xt generated by the moving average process as: xt = µ + εt + θ1εt−1, where εt are independently identically distributed random variables with E(εt) = 0, and V ar(εt) = σ2. Calculate the unconditional mean and the unconditional variance of xt. What is meant by saying that a process like xt is invertible? What condition would assure that xt is invertible? If θ = 0.75, does xt satisfy the invertibility condition? What shapes of the ACF and PACF functions do you expect for xt? Derive the first 4 autocorrelations for this process (τ1 up to τ4). Carefully write the equations for the 1, 2, 3 and 4 step ahead forecasts for xt.
- expand function f(z) into the laurent series at point z0 for given intermediate circle: f(z) = 1/((z-2)*(z-3)) a) z0 = 2; 0 < |z - 2| < 1 b) z0 = 2; |z - 2| > 1 c) z0 = ∞; |z| > 3expand function f(z) into the laurent series at point z0 for given intermediate circle: f(z) = 1/((z-3)*(z+1)); z0 = 3; 0Consider the following configuration of solar photovoltaic arrays consisting of crystalline silicon solar cells. There are two subsystems connected in parallel, each one containing two cells. In order for the system to function, at least one of the two parallel subsystems must work. Within each subsystem, the two cells are connected in series, so a subsystem will work only if all cells in the subsystem work. Consider a particular lifetime value to, and suppose we want to determine the probability that the system lifetime exceeds t0. Let Ai denote the event that the lifetime of cell i exceeds t0 (i = 1, 2, , 4). We assume that the Ai's are independent events (whether any particular cell lasts more than t0 hours has no bearing on whether or not any other cell does) and that P(Ai) = 0.6 for every i since the cells are identical. Using P(Ai) = 0.6, the probability that system lifetime exceeds t0 is easily seen to be 0.5904. To what value would 0.6 have to be changed in order to increase…