Question

Suppose Z^ - hZ1 A where Z1,A are independent random variables with mean 0 Part a) If σ-4, σ:1-1 and φ- 2? 0.3 , what is σ Part b) If σ 6 and φ-0.7, what is ơỈ in order that σ}i-Ž ?

Answer 1

Part a)

$\sigma^2_{Z_2} = Var[Z_2] = Var[\phi Z_1 + A] = \phi^2 Var[Z_1] + Var[A]$ (A, Z1 are independent random variables)

$\sigma^2_{Z_2} = \phi^2 \sigma^2_{Z_1}+ \sigma^2_A = 0.3^2 * 1 + 4 = 4.09$

Part b)

From part a),

$\sigma^2_{Z_2} = \phi^2 \sigma^2_{Z_1}+ \sigma^2_A$

For $\sigma^2_{Z_2} = \sigma^2_{Z_1}$ ,

$\sigma^2_{Z_1} = \phi^2 \sigma^2_{Z_1}+ \sigma^2_A$

$\Rightarrow \sigma^2_A = \sigma^2_{Z_1} - \phi^2 \sigma^2_{Z_1}$

$\Rightarrow \sigma^2_A =(1 - \phi^2) \sigma^2_{Z_1}$

$\Rightarrow \sigma^2_A =(1 - 0.7^2) * 6 = 3.06$