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Problem :

Let $\Omega:=[0,1]$

How to define two independent random variables, which describe coin toss on the above omega?

My idea:

$\Omega=[0,1]$

$\Sigma = \mathbb{B}([0,1])$

Consider the Lebesgue measure

1-reverse coin 2-obverse coin

$X(w)= \begin{cases} 0 &\text{when } w \in [0,1/2) \\ 1 &\text{when } w\in [1/2,1) \end{cases} $

$Y(w)= \begin{cases} 1 &\text{when } w \in [0,1/4) \cup [1/2,3/4) \\ 0 &\text{when } w\in [1/4,1/2)\cup [3/4,1) \end{cases} $

It seems, that $P(X=1)=P(X=0)=P(Y=0)=P(Y=1)=1/2$

And, for example $P(X=0)P(Y=1)=1/4=P(X=0,Y=1)$

What do you think about this?

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Yes, your idea works just fine.

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