hwset5 - EE 278 Monday August 3 2009 Statistical Signal...

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

View Full Document Right Arrow Icon

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

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

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 discrete-time continuous-valued 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. Discrete-time Wiener process. Let { Z n : n ≥ } be a discrete-time 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....
View Full Document

{[ snackBarMessage ]}

Page1 / 5

hwset5 - EE 278 Monday August 3 2009 Statistical Signal...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online