Fundamentals

This page covers core probability concepts: axiomatic framework, random variables, moments, and limit theorems. For concrete laws, see Distributions.

A probability $$ on a space $$ satisfies:

1. $$ for every event $$
2. $$
3. $$-additivity: for disjoint events $$,

A probability space is a triple $$:

• $$: sample space
• $$: sigma-algebra of measurable events
• $$: probability measure

The sigma-algebra supports countable limits on events.

A random variable $$ is a measurable map $$ (or $$).

Its law is the pushforward measure $$.

• Discrete: mass $$
• Continuous: density $$ with $$

CDF: $$.

Expectation is the center of mass of a law:

Linearity: $$ (no independence required).

Transfer: $$.

For a sum: $$; if $$ and $$ are independent, the cross term vanishes.

The $$-th moment: $$. Central moments of order 3 and 4 define:

• Skewness: asymmetry of the distribution
• Kurtosis: tail heaviness relative to the normal

In finance, high kurtosis signals tail risk.

Conditional law of $$ given $$:

Independence: $$, or $$.

Independence $$ uncorrelatedness (except in the Gaussian case).

For i.i.d. $$ with $$:

The sample mean converges to the true mean — foundation of Monte Carlo and long backtests.

If $$ and $$:

Justifies widespread use of the normal distribution and approximation of estimation errors.