202405240846
Status: #idea
Tags: Probability Theory
State: #nascient

Sample Space

A sample space, typically denoted $\Omega$, represents the set of all possible outcomes of a random experiment.

For example, if I toss a coin once, a possible and natural sample space would be $\Omega =\{H,T\}$, but observe that the experiment does not force the sample space. As in multiple totally valid sample spaces exist for a given random experiment, and me the--data scientist--is the one that chooses what my sample space is.

If instead, I am interested in how many times the coin tumbles in the air, even though the random experiment is the same (tossing a coin once), the sample space would now be $\Omega =\{1,2,3,\dots \}=\mathbb{N}$.

Therefore, the sample space is defined both by the random experiment and the thing of interest.

The random of random experiment comes from the fact that the outcomes have an associated probability and that we cannot control which $\omega -$labelled marble we draw from our big bag called $\Omega$.

Finite Sample Spaces
Infinite Sample Spaces
Countable Sample Spaces
Uncountable Sample Spaces

References