0

If X and Y and Z are i.i.d. random variables then does this mean they each have the same mean and the same density distribution- if this is the case then they must be the same variable which means they must not be independent . Please explain.

user263904
  • 139
  • Having the same distribution does not make them the same variable. Why do you think that? Consider, for example, the probability space ${ 1,2 }$ where $\mathbb{P}$ is the uniform distribution, and the random variables $X(\omega)=1$ if $\omega=1$ and zero otherwise, and $Y(\omega)=1$ if $\omega=2$ and zero otherwise. Then $X$ and $Y$ have the same distribution (they are both Bernoulli(1/2)) but they are not the same variable. (In this example they are not independent either; in fact $Y=1-X$. Still, that's not the point of this little illustration.) – Ian Aug 26 '16 at 00:26

1 Answers1

1

Let $X, Y, Z$ be the results of tossing three coloured, but equally-biased, dice.

Clearly $X, Y, Z$ have the same distribution, but are not the same variable.   They may independently have three different values in any instance.

Graham Kemp
  • 129,094