Is it possible that two random variables have the same distribution and yet they are almost surely different?

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# Can two random variables have the same distribution, yet be almost surely different?

### 2 Answers

Let $X\sim N(0,1)$ and define $Y=-X$. It is easy to prove that $Y\sim N(0,1)$.

But $$ P\{\omega : X(\omega)=Y(\omega)\} = P\{\omega : X(\omega)=0,Y(\omega)=0\} \leq P\{\omega : X(\omega)=0\} = 0 \, . $$

Hence, $X$ and $Y$ are different with probability one.

Any pair of independent random variables $X$ and $Y$ having the same continuous distribution provides a counterexample.

In fact, two random variables having the same distribution are not even necessarily defined on the same probability space, hence the question makes no sense in general.

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Let $X\sim N(0,1)$ and define $Y=-X$. It is easy to prove that $Y\sim N(0,1)$.

But $$ P\{\omega : X(\omega)=Y(\omega)\} = P\{\omega : X(\omega)=0,Y(\omega)=0\} \leq P\{\omega : X(\omega)=0\} = 0 \, . $$

Hence, $X$ and $Y$ are different with probability one.

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