Is a collection of random variables always a random vector? Let $X_1, \ldots, X_n$ be a collection of $n$ random variables with the same sample space $\Omega$, the same $\sigma$-algebra $\mathcal{F}$ but not identically distributed, i.e., $P(X_1 = \omega)$ is not necessarily the same as say $P(X_2 = \omega)$.
Wikipedia defines a random vector as

[...] a column vector $\mathbf{X} = (X_1, \ldots, X_n)^T$ [...] whose
  components are scalar-valued random variables on the same probability
  space $(\Omega, \mathcal{F}, P)$ [...].

My question is whether the collection of $n$ random variables described in the first paragraph is a random vector or not. If not, how would you call it? An indexed family of random variables perhaps?
 A: Let us first recall the definition of random variable $X$ defined on probability space $(\Omega,\mathcal F,P)$ and with values from some measurable space $(S,\mathcal S)$; it is $(\mathcal F, \mathcal S)$-measurable function $$X: \Omega \to S$$
For definition of measurable function, see, for example, this wikipedia article.
The wikipedia page you quote say that random vector $X=(X_1,\ldots,X_n)$ is function $$X: \Omega \to S^n$$ where $X_1,\ldots,X_n$ are random random variables defined on the same probability space $(\Omega,\mathcal F,P)$ with values from measurable space $(S,\mathcal S)$, defined as $$X(\omega)=(X_1(\omega),\ldots,X_n(\omega)).$$
This definition doesn't even mention the distributions of $X_1,\ldots,X_n$; in particular, it does not say that they have to be the same. Therefore, the function above is a random vector for any random variables $X_1,\ldots,X_n$.
Actually, random vector $X$ is simply any random variable taking values from measurable space $(S^n,\mathcal S^n)$, where $\mathcal S^n$ is product $\sigma$-algebra, since $(\mathcal F,\mathcal S)$-measurability of it's components $X_1,\ldots,X_n$ is necessary and sufficient condition for $(\mathcal F,\mathcal S^n)$-measurability of $(X_1,\ldots,X_n)$.
