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.

**Contents**hide

#### 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 $Binmathcal{B}$ have probabilities, $mathbb{P}(B)$. When picking $omegainOmega$, all sets $B$ such that $omegain B$ take place, while all those such that $omeganotin B$ 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.