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- Suppose that {u: t = ...,-2,-1,0,1,2,...} is an independent time series with mean zero, variance σ²= 9.0. Suppose that the time series {x+: t = ...,-2,-1,0,1,2,...} satisfies the equation: 2 X= 1.5 X-1 - .5 X-2 +3.0+u- U-1- 3 a) Identify the time series. b) c) d) i) Determine the autocovariance function and autocorrelation function of the time series X+ - Xt-1- Find the random shock form of the time series. Suppose that the first five observations of the time series are x₁ = 2.5, x2 = 4.0, X3 = 3.7, x4 =1.0 and x5 1.5. = Use these observations to compute prediction intervals for the next 5 observations. (Compute both 95% and 66.7% prediction limits) ii) If the sixth observation turns out to be x6 = 4.3 use this information to re-compute prediction intervals for the next 4 observations. (Compute both 95% and 66.7% prediction limits)An efficiency study of the first shift (from 8:00 A.M. to 5:00 P.M.) at a certain factory indicates that an average worker who arrives on the job at 8:00 A.M. will have assembled Q(t) = -2t3 + 8t2 + 10t MPEG Players t hours later. How many hours after the start of the shift is the worker performing most efficiently?A time-series is a vector used to describe a process evolving as a function of time. Suppose c₁ € R³65 and C₂ € R³65 contains the number of cases of a virus over 365 consecutive days. Provide interpretations of the following mathematical statements (1 sentence per response). Interpretations should be in terms of virus cases, and not a re-statement of the mathematical expression. (a) (C₁)k+1 > (C₁)k for k = 1, 2, ..., 364. (b) c ₂ = 0 (c) ||c₁||1 > ||C₂||1, where || ||1 denotes the 1-norm.
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- Consider an infinite-server queueing system to which customers arrive according to a PP() (A > 0). Each customer's service time is exponentially distributed with rate u (u > 0). Since there are infinitely many servers, all customers begin their service immediately upon arrival (i.e., there is no queueing delay). Let X(t) be the number of customers in the system at time t. (a) Derive the limiting distribution (p) of {X(t) : t > 0}.If X:B(n, p), i.e., Binomial with n and p, a) Find the moment generating function of 4-5X. b) Use the moment generating function to find the expected value of X. c) If Y = 3X –5 find E(Y).Find the PSD of a random process X(t) if E[X(t)] =1 and R(T) = 1 + e-aIt, X,