h5-6 - Statistics 5101 Fall 2010 Homework Assignment 5 6...

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Statistics 5101, Fall 2010: Homework Assignment 5 & 6 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. Suppose A and B are independent events. Show that if A and B are also mutually exclusive events, then either Pr( A ) = 0 or Pr( B ) = 0. 5-4. Suppose A and B are independent events, and Pr( A ) = Pr( B ) = p . Calculate Pr( A B ).
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