随机算法 (Fall 2011)/Problem set 4: Difference between revisions
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== Problem 1== | == Problem 1== | ||
Let <math>p</math> and <math>q</math> be two probability distributions over <math>\Omega</math>. | Let <math>p</math> and <math>q</math> be two probability distributions over <math>\Omega</math>. | ||
* Give an explicit construction of a coupling <math>\mu</math> of <math>p</math> and <math>q</math> such that <math>\Pr_{(X,Y)\sim\mu}[X\neq Y]=\|p-q\|_{TV}</math>. | * Give an explicit construction of a coupling <math>\mu</math> of <math>p</math> and <math>q</math> such that | ||
::<math>\Pr_{(X,Y)\sim\mu}[X\neq Y]=\|p-q\|_{TV}</math>. | |||
Revision as of 06:35, 28 November 2011
Problem 1
Let [math]\displaystyle{ p }[/math] and [math]\displaystyle{ q }[/math] be two probability distributions over [math]\displaystyle{ \Omega }[/math].
- Give an explicit construction of a coupling [math]\displaystyle{ \mu }[/math] of [math]\displaystyle{ p }[/math] and [math]\displaystyle{ q }[/math] such that
- [math]\displaystyle{ \Pr_{(X,Y)\sim\mu}[X\neq Y]=\|p-q\|_{TV} }[/math].