{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

mid-term

# mid-term - STA 4005A Mid-term Examination(Total Marks 35...

This preview shows pages 1–2. Sign up to view the full content.

STA 4005A Mid-term Examination (Total Marks 35) October 25, 2007 Answer ALL questions: Time allowed: 1 1/2 hours. Let a t NID (0 , σ 2 a ). 1. Consider a MA(2) model: Z t = a t - 1 . 3 a t - 1 - 0 . 4 a t - 2 , σ 2 a = 5 . (a) (2 marks) Find E ( Z t ) and V ar ( Z t ). (b) (3 marks) Find the autocovariance function γ k , k = 1 , 2 , 3 , ... . (c) (2 marks) Find the autocorrelations ρ 1 and ρ 2 . 2. Consider the process Z t = 6 + X t where X t = 1 when t is odd and X t = 0 when t is even. (a) (3 marks) Find the joint distribution function of Z 1 , Z 2 and Z 3 . (b) (2 marks) Is { Z t } strictly stationary? If the answer is YES, prove it. If the answer is NO, explain why. 3. Suppose that a random process { Z t } ( t = 0 , ± 1 , ± 2 , ... ) is defined by Z t - αZ t - 1 = Y t - βY t - 2 where α, β are real constants and { Y t } is a process of zero mean uncorrelated random variables with variance σ 2 Y = 4. (a) (3 marks) Find the values of π i , i = 0 , 1 , 2 , 3 , ... if the model can be rewrit- ten as Z t = π 0 Y t + π 1 Y t - 1 + π 2 Y t - 2 + ... . (b) (2 marks) If α = 0 . 9 , β = 0 . 2 find the Cov ( Z t , Z t - 1 ). 4. (a) (3 marks) Suppose X t = 3 + Y t where the autocovariances of { Y t } are given by γ k = (0 . 5) k , k = 0 , 1 , 2 , 3 , ... . Find V ar ( X 4 + 2 X 5 + X 6 ). (b) (3 marks) Consider the model Z t = a t - 0 . 3 a t - 1 - 0 . 6 a t - 4 + 0 . 18 a t - 5 . Find the autocovariance function

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern