Construction of coupling and maximal coupling (probability theory)

[TaPaKaH]

Homework Statement

Let U, V be random variables on $[0,+\infty)$ with probability density functions $f_U(x)=2e^{-2x}$ and $f_V(x)=e^{-x}$.
1. Give a coupling of U and V under which $\{U\geq V\}$ with probability 1.
2. Give a maximal coupling of U and V.

Homework Equations

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

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.

 

[Ray Vickson]

Homework Statement

Let U, V be random variables on $[0,+\infty)$ with probability density functions $f_U(x)=2e^{-2x}$ and $f_V(x)=e^{-x}$.
1. Give a coupling of U and V under which $\{U\geq V\}$ with probability 1.
2. Give a maximal coupling of U and V.

Homework Equations

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

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.

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?

 

[TaPaKaH]

What do you mean by a "coupling" of U and V? Do you mean a bivariate density function with marginals $f_U, f_V$...

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.