{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Assignment 5 - Stat 5101(Geyer Fall 2009 Homework...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Stat 5101 (Geyer) Fall 2009 Homework Assignment 5 Due Wednesday, October 21, 2009 Solve each problem. Explain your reasoning. No credit for answers with no explanation. If the problem is a proof, then you need words as well as formulas. Explain why your formulas follow one from another. 5-1. Suppose X 1 , X 2 , . . . are IID random variables having mean μ and variance τ 2 . For each i 1 define Y i = 5 X j =1 X i + j Then Y 1 , Y 2 , . . . is called a moving average of order 5 time series, MA(5) for short. It is a weakly stationary time series. (So far this repeats the setup for Problem 4-6.) If Y n = 1 n n X i =1 Y i show that Y n - 5 μ = O p ( n - 1 / 2 ) . 5-2. Suppose that X 1 , X 2 , . . . are IID random variables. For some subset A of the real numbers, define Y i = I A ( X i ) , i = 1 , 2 , . . . and Y n = 1 n n X i =1 Y i . Show that Y n converges in probability to Pr( X i A ), and in fact Y n - Pr( X i A ) = O p ( n - 1 / 2 ) . Also show that Y n is the fraction of X 1 , . . . , X n that lie in A . 5-3. Define Ω = { 1 , 2 , . . . , 10 } A = { ω Ω : ω < 5 } B = { ω Ω : 2 < ω < 8 } C = { 10 } 1
Background image of page 1

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

View Full Document Right Arrow Icon
(a) Calculate A c where complements are taken relative to Ω.
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}