Stat150_Spring08_homework2

# Stat150_Spring08_homework2 - Stochastic Processes Homework...

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Unformatted text preview: Stochastic Processes Homework 2: Wednesday, January 30 Due: Wednesday, February 6 The total variation distance between two continuous real-valued random variables X and Y with densities f X , f Y is denoted d TV ( X, Y ) and defined as d TV ( X, Y ) = Z R | f X ( z )- f Y ( z ) | dz. Illustration of total variation distance: the total area shaded light grey is the total variation distance between random vari- ables having densitites f X and f Y . f_X f_Y 1. Show that the total variation distance between two continuous real-valued random variables X, Y satisfies d TV ( X, Y ) = 2 sup A ⊆ R | P { X ∈ A } - P { Y ∈ A }| . You may want to prove and use the fact that the two “halves” of the light grey region in the figure above have the same area. 2. Let X and Y be two continuous real-valued random variables X, Y defined on the same probability space. Using (1) , show that P { X = Y } ≤ 1- 1 2 d TV ( X, Y ). Note that the right-hand side does not depend on the joint distribution of...
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## This note was uploaded on 05/07/2008 for the course STAT 150 taught by Professor Evans during the Spring '08 term at Berkeley.

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