Drawing independent Random Variables out of a Probability Distribution

Question

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.

Answer 1

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.

Answer 2

The motivation for my questions stems from a phrase of MIT professor Tsitsiklis: "We then draw independent random variables out of this distribution so that these Xi's are independent and identically distributed, i.i.d. for short. What's going on here is that we're carrying out a long experiment during which all of these random variables are drawn."

To better understand this statement we might note that a Probability Distribution can be associated with experiments that generate outcomes governed by the Probabilistic Model defined by the Probability Distribution. An example is the toss of a coin. Each toss can be conceptualized as an experiment governed by a Probability Law defined by the Bernoulli Distribution. We can define Random Variables that capture the possible outcome of each of those experiments (tosses). We then have a sequence of Random Variables associated with the same (Bernoulli) Distribution.

As a special case, we may consider a probabilistic model in which we repeat independently many times the same experiment. There's a certain event A associated with that experiment that has a certain probability, and each time that we carry out the experiment, we use an indicator variable to indicate whether the outcome was inside the event or outside the event. So Xi is 1 if A occurs, and it is 0 otherwise. In sum then, the sequence of n experiments generates a sequence of n Random Variables, each associated with the same Probabilistic Model / Probability Distribution.

Since we do not know in advance the outcome of each unique experiment, Xi is not a real value but a Random Variable that can take the values 0 and 1 with a certain probability.

Perhaps it is in that sense, that the term 'drawn out of the distribution'is used, in reference to a sequence of Random Variables.