1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Construction of coupling and maximal coupling (probability theory)

  1. Jan 19, 2013 #1
    1. The problem statement, all variables and given/known data
    Let U, V be random variables on [itex][0,+\infty)[/itex] with probability density functions [itex]f_U(x)=2e^{-2x}[/itex] and [itex]f_V(x)=e^{-x}[/itex].
    1. Give a coupling of U and V under which [itex]\{U\geq V\}[/itex] with probability 1.
    2. Give a maximal coupling of U and V.

    2. Relevant equations
    Cumulative distribution functions (probability measures) for U and V are:
    [itex]P_U([a,b])=P(u\in[a,b])=e^{-2a}-e^{-2b}[/itex],
    [itex]P_V([a,b])=P(v\in[a,b])=e^{-a}-e^{-b}[/itex].

    3. The attempt at a solution
    I'm having difficulties constructing coupling required in exercise 1. I tried introducing third variable W and letting U=max{V,W}, U=V+W, U=V*W and U=(V+W)/2 but in none of the four cases I could come up with a 'good' cumulative distribution function for W.
     
    Last edited: Jan 19, 2013
  2. jcsd
  3. Jan 19, 2013 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    What do you mean by a "coupling" of U and V? Do you mean a bivariate density function with marginals ##f_U, f_V##, or do you mean a probability space ##\Omega## in which ##U## and ##V## are functions ##U(\omega), V(\omega)##? And: what do you mean by a "maximal" coupling?
     
  4. Jan 19, 2013 #3
    Yes, I need a random variable ##(\hat{u},\hat{v})## on ##[0,+\infty)\times[0,+\infty)## such that ##u\overset{D}{=}\hat{u}## and ##v\overset{D}{=}\hat{v}##.
    Easy case is to put ##\hat{u}## and ##\hat{v}## independent with marginals above, second example I thought of is ##\hat{u}=\min\{\hat{v},w\}## with ##\hat{v}## and w being independent 'copies' of v, but this gives the opposite of needed ##\hat{\mathbb{P}}(\{\hat{u}\geq\hat{v}\})=1## (might as well be a typo in the exercise).

    Maximal coupling is such ##(\hat{u},\hat{v})## that the total variation norm ##\|\mathbb{P}_U-\mathbb{P}_V\|_{tv}=2\hat{\mathbb{P}}(\{\hat{u} \neq \hat{v}\})## while ##u\overset{D}{=}\hat{u}## and ##v\overset{D}{=}\hat{v}##.
    The total variation norm is 1/2, computed by hand.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Construction of coupling and maximal coupling (probability theory)
  1. Coupled ODE's (Replies: 3)

Loading...