It might sound weird but up until now in my studies the randomness was assigned to a Cumulative distribution function which is a deterministic mapping (for discrete and continuous) random variables.
My question is can we go a little more further ? i.e. are there random phenomenon (maybe from physics or economy or constructed by maths) to which it is proven that we cannot assign a (deterministic) Cumulative distribution function ? Thank you for your answer,
To elaborate on @fesman's comment, a random variable $X$ is a measurable function on a measurable space $(\Omega, \mathbb P)$. We define for any such $X$ the cumulative distribution function $F_X:(-\infty,\infty)\to[0,1]$ by $$ F_X(x) := \mathbb P(X\leqslant x), $$ or to be more explicit, $$ F_X(x) := \mathbb P(\{\omega\in\Omega : X(\omega)\leqslant x). $$ We cannot define a "random phenomenon" without a distribution function. It is more likely that the "randomness" in your studies did not consist of rigorous probability theory.