I am currently reading notes for a probability class I am taking and am having trouble with notation.
Let Ω = [0, 1], and let P be the uniform probability measure on Ω so that P([a, b]) = b − a
There are two notations for a sequence of random variables:
$X_n$ ~ $n \cdot Bernoulli(\frac{1}{n})$ and $X_n (\omega)$ = $n \cdot \mathbb{1}_\omega \in [0, \frac{1}{n}]$. I was told that notation one and two are different, as the second notation means that the $X_n$'s are not independent. I am not really sure what this means. Could someone please help me clarify the difference between these two notations?
Is the reason $X_n$'s aren't independent for equation 2 because the $\omega$ value we have for each of the $X_n$'s are the same? So for example if $\omega$ = $\frac{1}{2}$ for $X_1$, we have that $X_1 (\omega)$ = $1$. Then this means that $X_2 (\omega)$ = $2$?
Any help would be appreciated, thank you!