Assume that wt ∼ wn(0, σ2 ), t = 1, 2, . . . ,. Define...

Assume that wt ∼ wn(0, σ2 ), t = 1, 2, . . . ,. Define Yt = µ + wt − θwt−1, where µ and θ are constants. Find E(Yt), var(Yt), and cov(Yt , Yt+k), where k is an integer and k ≥ 0

Expert Solution

Assume that wt ∼ wn(0, σ2 ), t = 1, 2, . . . ,. Define Yt = µ + wt − θwt−1, where µ and θ are constants. Find E(Yt), var(Yt), and cov(Yt , Yt+k), where k is an integer and k ≥ 0

Colby Messinger answered 2 years ago

Consider the time series Xt = 4t + Wt + 0.9Wt−1, where Wt ∼ N(0, σ2...

Consider the time series Xt = 4t + Wt + 0.9Wt−1, where Wt ∼ N(0, σ2 ). (i)What are the mean function and the variance function of this time series? Is this time series stationary? Justify your answer (ii). Consider the first differences of the time series above, that is, consider Yt = Xt − Xt−1. What are the mean function and autocovariance function of this time series? Is this time series stationary? Justify your answer

1. Let W1, . . . , Wn be i.i.d. N(µ, σ2). A 100(1 − α)%...

1. Let W1, . . . , Wn be i.i.d. N(µ, σ2). A 100(1 − α)% confidence interval for σ2 is ( ([n − 1]*S2 ) / (χ21−α/2,n−1 ) , ([n − 1]S2) / ( χ2α/2,n−1) , where χ2u,k denotes the 100×uth percentile of the χ2 distribution with k degrees of freedom, S2 = (n − 1)−1 * the sum of (Wi − Wbar )2 (from 1 to n) is the sample variance and Wbar = n−1 * the sum...

y′ = t, y(0) = 1, solution: y(t) = 1+t2/2 y′ = 2(t + 1)y, y(0)...

y′ = t, y(0) = 1, solution: y(t) = 1+t2/2 y′ = 2(t + 1)y, y(0) = 1, solution: y(t) = et2+2t y′ = 5t4y, y(0) = 1, solution: y(t) = et5 y′ = t3/y2, y(0) = 1, solution: y(t) = (3t4/4 + 1)1/3 For the IVPs above, make a log-log plot of the error of Backward Euler and Implicit Trapezoidal Method, at t = 1 as a function of hwithh=0.1×2−k for0≤k≤5.

y′ = t, y(0) = 1, solution: y(t) = 1+t2/2 y′ = 2(t + 1)y, y(0)...

y′ = t, y(0) = 1, solution: y(t) = 1+t2/2 y′ = 2(t + 1)y, y(0) = 1, solution: y(t) = et2+2t y′ = 5t4y, y(0) = 1, solution: y(t) = et5 y′ = t3/y2, y(0) = 1, solution: y(t) = (3t4/4 + 1)1/3 For the IVPs above, make a log-log plot of the error of Explicit Trapezoidal Rule at t = 1 as a function ofhwithh=0.1×2−k for0≤k≤5.

6. The function f(t) = 0 for − 2 ≤ t < −1 −1 for −...

6. The function f(t) = 0 for − 2 ≤ t < −1 −1 for − 1 ≤ t < 0 0 for t = 0 1 for 0 ≤ t < 1 0 for 1 ≤ t ≤ 2 can be extended to be periodic of period 4. (a) Is the extended function even, odd, or neither? (b) Find the Fourier Series of the extended function.(Just write the final solution.)

Let X = [1, 0, 2, 0]tand Y = [1, −1, 0, 2]t. (a) Find a...

Let X = [1, 0, 2, 0]tand Y = [1, −1, 0, 2]t. (a) Find a system of two equations in four unknowns whose solution set is spanned by X and Y. (b) Find a system of three equations in four unknowns whose solution set is spanned by X and Y. (c) Find a system of four equations in four unknowns that has the set of vectors of the form Z + aX + bY as its general solution where...

y' = 2 + t^2 + y^2 0<t<1 y(0)=0 use the euler method to determine step...

y' = 2 + t^2 + y^2 0<t<1 y(0)=0 use the euler method to determine step size (h) to keep global truncation error below .0001

Solve the initial value problem: Y''-4y'+4y=f(t) y(0)=-2, y'(0)=1 where f(t) { t if 0<=t<3 , t+2...

Solve the initial value problem: Y''-4y'+4y=f(t) y(0)=-2, y'(0)=1 where f(t) { t if 0<=t<3 , t+2 if t>=3 }

X(t)=Atan(wt)+Bcot(wt) and Y(t)=Btan(wt)+Acot(wt) both are random process, w is a constant, A and B are zero-mean...

X(t)=Atan(wt)+Bcot(wt) and Y(t)=Btan(wt)+Acot(wt) both are random process, w is a constant, A and B are zero-mean and variance of A and B is σ ^2, they are independent random variables, a-) find autocorrelation function of X(t) and Y(t) b-) find the cross correlation function of X(t) and Y(t)

1. Express the function f(t) = 0, -π/2<t<π/2 &nbsp

1. Express the function f(t) = 0, -π/2<t<π/2 = 1, -π<t<-π/2 and π/2<t<π with f(t+2π)=f(t), as a Fourier series.

Question

Assume that wt ∼ wn(0, σ2 ), t = 1, 2, . . . ,. Define...

Solutions

Expert Solution

Related Solutions

Consider the time series Xt = 4t + Wt + 0.9Wt−1, where Wt ∼ N(0, σ2...

1. Let W1, . . . , Wn be i.i.d. N(µ, σ2). A 100(1 − α)%...

y′ = t, y(0) = 1, solution: y(t) = 1+t2/2 y′ = 2(t + 1)y, y(0)...

y′ = t, y(0) = 1, solution: y(t) = 1+t2/2 y′ = 2(t + 1)y, y(0)...

6. The function f(t) = 0 for − 2 ≤ t < −1 −1 for −...

Let X = [1, 0, 2, 0]tand Y = [1, −1, 0, 2]t. (a) Find a...

y' = 2 + t^2 + y^2 0<t<1 y(0)=0 use the euler method to determine step...

Solve the initial value problem: Y''-4y'+4y=f(t) y(0)=-2, y'(0)=1 where f(t) { t if 0<=t<3 , t+2...

X(t)=Atan(wt)+Bcot(wt) and Y(t)=Btan(wt)+Acot(wt) both are random process, w is a constant, A and B are zero-mean...

1. Express the function f(t) = 0, -π/2<t<π/2 &nbsp