(5 points) Consider the series xt = sin(2nUt), t = 1.2, . . . , where...
(a) Let X(t) sin t, where is uniformly distributed over the interval [0, 27]. Verify (i) The discrete-time process (t), t = 1,2,... is weakly, but not strongly stationary (ii) The continuous-time stochastic process X(t), t> 0 is neither weakly nor strongly stationary (a) Let X(t) sin t, where is uniformly distributed over the interval [0, 27]. Verify (i) The discrete-time process (t), t = 1,2,... is weakly, but not strongly stationary (ii) The continuous-time stochastic process X(t), t> 0...
Consider the following model on a return series rt=t+ at +0.25at-1, where at riid N(0,02), t = 1, ... ,T. (a) What are the mean function and autocovariance function for this return series? Is this return series {rt} weakly stationary? Justify your answer. (b) Consider first differences of the return series above, that is, consider wt = Vrt=rt – Pt-1. What are the mean function and autocovariance function for this time series? Is this time series {wt} weakly stationary? Justify...
Consider the time series Consider the time series of Xt Xt-1 + Wt, where Wt ~ N(0, σ2) and Xi. Derive the general form for var(X). (Hint: i=1 5 ? -n(n+1)(2n+1)
2. Consider the time series X, = 2 + 0.5t +0.8X1-1 + W, where W N(0.1). (a) (8 points) Calculate E(X2) Is this process weakly stationary? Give reasons for your answer. Hint: Find the mean function of {X) and then substitute t = 20. (b) (3 points) Calculate Var(X20) Question 2 continues on the next page... Page 4 of 12 c)(4 points) Consider the first differences of the time series above, that is Is {%) a weakly stationary process. Prove...
Let Xn = a sin(bn+Z), where n ∈ Z, a, b ∈ [0, ∞) are constant, and Z has a continuous uniform distribution on [−π, π] (i.e. Z ∼ U([−π, π])). Show that Xn is stationary. (Hint: sin(x) sin(y) = 1 2 (cos(x − y) − cos(x + y)) may be helpful). l. Let Xn-a sin(bn+ Z), where n є z, a, b є lo,00) are constant, and Z has a continuous uniform distribution on [-π, π] (i.e. Z ~...
4. consider a random process Xt Sin(TOOt+ φ). / nen/ent a 2π (a) Consider φ ~ Uniform(-π, π). Plot the process for t E (0, 1000) (b) Consider φ ~ Uniform(0,π), plot the process for t E (0, 1000)
Is (20 points) The complex exponential Fourier series of a signal xt) over 0<t<T is given as shown below. icos nas x(t)= (a) Calculate the period T (b) Determine the average value of x(1) (C) Find the amplitude of the fifth harmonic,
The sample data x1,x2,...,xn sometimes represents a time series, where xt = the observed value of a response variable x at time t. Often the observed series shows a great deal of random variation, which makes it difficult to study longer-term behavior. In such situations, it is desirable to produce a smoothed version of the series. One technique for doing so involves exponential smoothing. The value of a smoothing constant α is chosen (0 < α < 1). Then with...
(4) (a) Compute the Fourier series for the function f(s)-- interval [-T, on the (b) Compute the solution u(t,a) for the partial differential equation on the interval [o, ) luWith u(t, 0) u(t,1)-0 for t>0 (boundary conditions) u(o,z)-3 sin(2x)-5 sin(5z) + sin(6z), for O < < 1 (initial conditions) (20 points) (4) (a) Compute the Fourier series for the function f(s)-- interval [-T, on the (b) Compute the solution u(t,a) for the partial differential equation on the interval [o, )...
(4) (a) Compute the Fourier series for the function f(x) interval [-π, π]. 1-z on the (b) Compute the solution u(t, z) for the partial differential equation on the interval [0, T): 16ut = uzz with u(t, 0)-u(t, 1) 0 for t>0 (boundary conditions) (0,) 3 sin(2a) 5 sin(5x) +sin(6x). for 0 K <1 (initial conditions) (20 points) Remember to show your work. Good luck. (4) (a) Compute the Fourier series for the function f(x) interval [-π, π]. 1-z on...