Question

Let {Z_t} be a sequence of independent normal random variables, each with mean 0 and variance ?^2 and let a, b and c be constants. Which, if any, of the following processes are stationary? For each stationary process specify the mean, the variance and autocovariance function. (a) X_t = a + bZ_t + cZ_{t-2} (b) X_t = Z_1 cos(ct) + Z_2 sin(ct) (c) X_t = Z_t cos(ct) + Z_{t-1} sin(ct) (d) X_t = a + bZ_0 (e) X_t = Z_0 cos(ct)

          Let {Z_t} be a sequence of independent normal random variables, each with mean 0 and variance ?^2 and let a, b and c be constants. Which, if any, of the following processes are stationary? For each stationary process specify the mean, the variance and autocovariance function.

(a) X_t = a + bZ_t + cZ_{t-2}
(b) X_t = Z_1 cos(ct) + Z_2 sin(ct)
(c) X_t = Z_t cos(ct) + Z_{t-1} sin(ct)
(d) X_t = a + bZ_0
(e) X_t = Z_0 cos(ct)

Let Zt be a sequence of independent normal random variables, each with mean 0 and variance ?^2 and let a, b and c be constants. Which, if any, of the following processes are stationary? For each stationary process specify the mean, the variance and autocovariance function.

(a) Xt = a + bZt + cZt-2
(b) Xt = Z1 cos(ct) + Z2 sin(ct)
(c) Xt = Zt cos(ct) + Zt-1 sin(ct)
(d) Xt = a + bZ0
(e) Xt = Z0 cos(ct)

Added by Michael R.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Areen Dabadghav

02/28/2023

Step 1

The mean, variance, and autocovariance function are as follows: Mean: E(Xt) = a + bE(Zt) + cE(Zt-2) = a Variance: Var(Xt) = b^2Var(Zt) + c^2Var(Zt-2) = b^2σ^2 + c^2σ^2 Autocovariance function: Cov(Xt, Xt-k) = b^2Cov(Zt, Zt-k) + c^2Cov(Zt-2, Zt-k-2) = 0 (for k ≠ Show more…

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Let {Zt} be a sequence of independent normal random variables, each with mean 0 and variance ̃σ^2, and let a, b and c be constants. Which, if any, of the following processes are stationary? For each stationary process specify the mean, the variance and autocovariance function. (a) Xt = a + bZt + cZt-2 (b) Xt = Z1 cos(ct) + Z2 sin(ct) (c) Xt = Zt cos(ct) + Zt-1 sin(ct) (d) Xt = a + bZ0 (e) Xt = Z0 cos(ct)