Stat150_Spring08_homework2

Stat150_Spring08_homework2 - Stochastic Processes Homework...

Info iconThis preview shows page 1. Sign up to view the full content.

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: 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...
View Full Document

Ask a homework question - tutors are online