I have a difficulty grasping the intuitive meaning of drawing independent Random Variable out of a Probability Distribution. A probability distribution assigns probabilities to discrete outcomes or probability densities to continuous outcomes commonly represented by a Random Variable.
Therefore it seems that when I draw something out of a Probability Distribution, what I end up having in my hands is an Event comprised from possible outcomes along with a probability assigned to that Event by the PMF or PDF.
In what sense then, I draw Random Variables out of a Probability Distribution?
Your advice will be appreciated.
Best Answer
Drawing a random variable $X$ with realisation $x$ is like picking a point $\omega$ in a certain space $\Omega$ (this can be formalised as being equivalent). On that space $\Omega$, a collection of sets $\mathcal{B}$ can be constructed so that all sets $B\in\mathcal{B}$ have probabilities, $\mathbb{P}(B)$. When picking $\omega\in\Omega$, all sets $B$ such that $\omega\in B$ take place, while all those such that $\omega\not\in B$ do not take place. Events thus occur according to whether or not the realised value $\omega$ is within the corresponding set $B$. Therefore, to answer your question more specifically, it is the opposite: events are not drawn from the probability distribution, only outcomes, for which all possible events occur or do not occur.