For each of the time series models below undertake the following steps: • Write each time series in characteristic polynomial form using backshift operator (MA or AR or ARMA polynomials as required) • Factorise the relevant polynomials and find the roots of the polynomials. • You are given the following additional information regarding the relationship between stationarity and the characteristic polynomial roots. An AR(p) process is a stationary process if and only if the modulus of all the roots of the AR characteristic polynomial are greater than one. An MA(q) process is always stationary if the MA coefficients are finite. - An ARMA (p,q) model admits a unique stationary solution when the AR characteristic poly- nomial and the MA characteristic polynomial have no common roots and the AR polynomial satisfies that it has no unit roots. ● Use this information to determine which of the below models is stationary. (Make sure to justify your answer.) (c) Yt+1+ Yt - (1 − B)(1 + B)Yt+1 = Vet

For each of the time series models below undertake the following steps: Write each time series in characteristic polynomial form using backshift operator (MA or AR or ARMA polynomials as required) Factorise the relevant polynomials and find the roots of the polynomials. • You are given the following additional information regarding the relationship between stationarity and the characteristic polynomial roots. ● ● An AR(p) process is a stationary process if and only if the modulus of all the roots of the AR characteristic polynomial are greater than one. An MA(q) process is always stationary if the MA coefficients are finite. An ARMA (p,q) model admits a unique stationary solution when the AR characteristic poly- nomial and the MA characteristic polynomial have no common roots and the AR polynomial satisfies that it has no unit roots. ● Use this information to determine which of the below models is stationary. (Make sure to justify your answer.) (e) Let {Y} be a stationary process with…

For each of the time series models below undertake the following steps: • Write each time series in characteristic polynomial form using backshift operator (MA or AR or ARMA polynomials as required) • Factorise the relevant polynomials and find the roots of the polynomials. • You are given the following additional information regarding the relationship between stationarity and the characteristic polynomial roots. An AR(p) process is a stationary process if and only if the modulus of all the roots of the AR characteristic polynomial are greater than one. An MA(q) process is always stationary if the MA coefficients are finite. An ARMA (p,q) model admits a unique stationary solution when the AR characteristic poly- nomial and the MA characteristic polynomial have no common roots and the AR polynomial satisfies that it has no unit roots. Use this information to determine which of the below models is stationary. (Make sure to justify your answer.) (a) VY + ²3 (Yt + 5Y₁-1) = € - €t-1 − −2+ 3…

An engineer designs a circuit, seen below. In this system, if the first component fails, the functionality moves immediately to the second component and so on. The system only fails when the fifth component fails. Each component has a lifetime that is exponentially distributed with λ = 0.01 and components fail independently of one another. Define A, to be the length of time component i lasts. Let Y = the time at which the new system fails. Given 2 لنا 5 A-Exponential (0.01) Y=A1+A2+A3+A4+A5 1. Find the probability that this new system will last fewer than 50 hours.