2023-12-07

Trinity

The basic idea behind probability theory can be summarised in an outrageously compressed form in the following collection of symbols:

$$\left[\Omega,\mathcal{F},X^{-1},X_\mathbb{R},\mathbb{P}_{[0,1]}\right]$$

Where:

• $\Omega$ : The sample space.
• $\mathcal{F}$ : The $\sigma$-algebra on $\Omega$, defining measurable events.
• $X^{-1}$ : The pre-image operation under the random variable, illustrating how events are generated from the outcomes mapped by $X$.
• $X_{\mathbb{R}}$ : The random variable itself, mapping outcomes in $\Omega$ to real numbers in $\mathbb{R}$.
• $\mathbb{P}_{[0,1]}$ : The probability measure, assigning probabilities to events into the range of $[0, 1]$.

More commonly you will find: $$\left [ \Omega, \mathcal{F}, \mathbb{P} \right ]$$