Probability

Dependencies:

1. σ-algebra
2. Measure
3. σ-algebra is closed under countable intersections

Read sections 1.1, 1.2, 1.3 and 1.4 from [prob-and-rand-proc] for an intuitive understanding and to know the definitions of 'event', 'sample space', 'probability' and 'probability measure'. You may optionally read chapter 0 too.

A probability space is a triple $(\Omega, \mathcal{F}, P)$, where:

• $\Omega$ is the sample space, also called the set of all outcomes.
• $\mathcal{F}$ is a $\sigma$-algebra over $\Omega$. $\mathcal{F}$ is called the set of all events.
• $P: \mathcal{F} \mapsto [0, 1]$ is a measure over $(\Omega, \mathcal{F})$ such that $P(\Omega) = 1$. $P$ is called a probability measure.

One can usually let $\mathcal{F}$ be equal to the power-set of $\Omega$. But for certain infinite-sized $\Omega$, certain probability measures cannot be defined over the power-set of $\Omega$ (see the Banach-Tarski paradox or Vitali set).

Additional properties

These simple properties can be proven using the definition of probability space above:

• Let $S = \{A_1, A_2, \ldots\}$ be a countable set of events. Then $\bigcap_{A \in S} A \in \mathcal{F}$.
• Let $A$ be an event. Define $\overline{A} = \Omega - A$. Then $P(\overline{A}) = 1 - P(A)$ (since $A$ and $\overline{A}$ are disjoint and $A \cup \overline{A} = \Omega$).
• $A \subseteq B \implies P(A) \le P(B)$.
• $A \subseteq B \implies P(B-A) = P(B) - P(A)$ (because $A$ and $B-A$ are disjoint).
• $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.

References

Dependency for:

1. Linearity of expectation
2. Random variable
3. Independence of random variables (incomplete)
4. Conditional expectation
5. Expected value of a random variable
6. Conditional variance
7. Conditional probability (incomplete)
8. Independence of composite events

Info:

• Depth: 3
• Number of transitive dependencies: 5

Transitive dependencies:

1. /sets-and-relations/countable-set
2. /sets-and-relations/de-morgan-laws
3. σ-algebra
4. Measure
5. σ-algebra is closed under countable intersections