# 4.8.5 Exercises

###### Exercises
4.18

Exhibit 4.14 indicates a realization of 50 consecutive terms of a variance 1 Gaussian white noise.5

Exhibit 4.14: Realization of 50 consecutive terms of a variance 1 Gaussian white noise for use in Exercise 4.18 and subsequent exercises.

Use this to generate a corresponding realization of the MA(2) process

[4.59]

where tW is a variance 1 Gaussian white noise.

4.19

Use the white noise realization of Exhibit 4.14 to generate and graph a realization of the AR(2) process

[4.60]

where tW is a variance 1 Gaussian white noise. Initialize the realization with terms 0x = 1x= 0.

4.20

Use the white noise realization of Exhibit 4.14 to generate and graph a realization of the ARMA(1,1) process

[4.61]

where tW is a variance 1 Gaussian white noise. Initialize the realization with term 0x = 0.

4.21

Derive expressions for the unconditional and conditional means, E(tX) and t–1E(tX), of an MA(q) process [4.53].

4.22

Derive expressions for the unconditional and conditional means, E(tX) and t–1E(tX), of an AR(p) process [4.55], assuming they exist.