- #1

TaPaKaH

- 54

- 0

## Homework Statement

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.

## Homework 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].

## 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: