I am reading a book that in one page it talks about cdf of a random vector. This is from the book:
Given $X=(X_1,...,X_n)$, each of the random variables $X_1, ... ,X_n$ can be characterized from a probabilistic point of view by its cdf.
However the cdf of each coordinate of a random vector does not completely describe the probabilistic behaviour of the whole vector. For instance, if $U_1$ AND $U_2$ are two independent random variables with the same cdf $G(x)$, the vectors $X=(X_1, X_2)$ defined respectively by $X_1=U_1$, $X_2=U_2$ and $X_1=U_1$, $X_2=U_1$ have each of their coordinates with the same cdf, and they are quite different.
My question is:
From the very last paragraph, it says $U_1$ and $U_2$ are coming from the same c.d.f. And then they define $X=(X_1, X_2)$, but they say $X=(X_1, X_2)$ is different from $X=(X_1, X_1)$. I don't really understand why the two $X$ are different.
(i.e. I don't understand why $X=(X_1, X_2)$ and $X=(X_1, X_1)$ are different). Isn't $X_1$ the same as $X_2$, so it doesn't matter whether you put two $X_1$ to form $X=(X_1, X_1)$ or put one $X_1$ and one $X_2$ to form $X=(X_1, X_2)$. Shouldn't they be the same? why does the author says they are "quite different"?
Could someone explain why they are different?