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**1. Homework Statement**

Take Ω = [0, 1] and P the uniform probability.

(a) Give an example of two random variables X and Y deﬁned on Ω that

both have the same Bernoulli distribution with parameter θ = 1/3.

(b) Give an example of such random variables that are independent, and not

independent.

**2. Homework Equations**

For a Bernoulli distribution:

Pr(x=0)= 1-θ

Pr(x=1)= θ

**3. The Attempt at a Solution**

I have that Pr(X=0)=Pr(Y=0)=2/3

Pr(X=1)=Pr(Y=1)=1/3

What I am not sure about is the part where I am asked for the examples of 2 random variables with this distribution. Can I simply say let X represent the chance of rolling a 1 on a 3 sided dice as an example? Then what would I choose for Y such that X and Y are independent for the second part?

Thanks for your help.