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Unformatted text preview: EE 278 Monday, August 3, 2009 Statistical Signal Processing Handout #14 Homework #5 Due Wednesday, August 12, 2009 1. Absolute value random walk. Let X n be a random walk defined by X = 0 , X n = n summationdisplay i =1 Z i , n ≥ 1 , where { Z i } is an i.i.d. process with P( Z 1 = 1) = P( Z 1 = +1) = 1 2 . Define the absolute value random process Y n =  X n  . a. Find P { Y n = k } . b. Find P { max { Y i : 1 ≤ i < 20 } = 10  Y 20 = 0 } . 2. Random walk with random start. Let X be a random variable with pmf p X ( x ) = braceleftBigg 1 5 x ∈ { 2 , 1 , , +1 , +2 } otherwise Suppose that X is the starting position of a random walk { X n : n ≥ } defined by X n = X + n summationdisplay i =1 Z i , where { Z i } is an i.i.d. random process with P( Z 1 = 1) = P( Z 1 = +1) = 1 2 and every Z i is independent of X . a. Does X n have independent increments? Justify your answer. b. What is the conditional pmf of X given that X 11 = 2 ? 3. Markov processes. Let { X n } be a discretetime continuousvalued Markov random process, that is, f ( x n +1  x 1 , x 2 , . . ., x n ) = f ( x n +1  x n ) for every n ≥ 1 and for all sequences ( x 1 , x 2 , . . ., x n +1 ). a. Show that f ( x 1 , . . ., x n ) = f ( x 1 ) f ( x 2  x 1 ) ··· f ( x n  x n 1 ) = f ( x n ) f ( x n 1  x n ) ··· f ( x 1  x 2 ) . b. Show that f ( x n  x 1 , x 2 , . . ., x k ) = f ( x n  x k ) for every k such that 1 ≤ k < n . c. Show that f ( x n +1 , x n 1  x n ) = f ( x n +1  x n ) f ( x n 1  x n ), that is, the past and the future are independent given the present. 4. Discretetime Wiener process. Let { Z n : n ≥ } be a discretetime white Gaussian noise process; that is, Z 1 , Z 2 , Z 3 , . . . are i.i.d. N (0 , 1). Define the process { X n : n ≥ } by X = 0 and X n = X n 1 + Z n for n ≥ 1....
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 Spring '11
 Eggers
 Signal Processing

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