# Probability Review

Updated 2017-10-20

## Axioms of Probability

\(\Omega\) is the set of all possible outcomes. An event is a subset of \(\Omega\).

- Probability must be between 0 and 1: \(0\leq \mathbb P(A) \leq 1\) for any event \(A\)
- If events \(A\) and \(B\) are mutually exclusive (\(A \cap B=\emptyset\)) then \(\mathbb P(A\cup B)=\mathbb P(A) + \mathbb P(B)\)
- The probability of the sample space is \(\mathbb P(\Omega)=1\) (always will happen) similarly, the probability of the empty set is \(\mathbb P(\emptyset)=1\).

## Events

### Mutually Exclusive

Events \(A\) and \(B\) are __mutually exclusive__ if and only if \(\mathbb P(A\cup B)=\mathbb P(A)+\mathbb P(B)\).

### Independent

Events \(A\) and \(B\) are __independent__ if and only if \(\mathbb P(A\cap B)=\mathbb P(A)\mathbb P(B)\).

### Conditional Probability

Probability of event \(A\) given \(B\) is \(\mathbb P(A\vert B)=\frac{\mathbb P(A\cap B)}{\mathbb P(B)}\), \(\mathbb P(B)\neq 0\).

### Total Probability

For all mutually exclusive events \(B_i\) that **partitions** \(\Omega\), the probability of event \(A\) is \(\mathbb P(A)=\sum A\cap B_i\).

## Random Variables

**Random variable** maps outcomes \(\omega \in \Omega\) of a probabilistic experiment to a real number \(X(\omega)\in\mathbb R\)

### Cumulative Distribution Functions

\[F_X(x)=\mathbb P(X(\omega) \leq x) \quad\text{for any } x\in \mathbb R\]Properties:

- \[F_X(-\infty)=0, F(\infty)=1\]
- \[\mathbb P(X\in (a, b])=\mathbb P(a<X\leq b)=F_X(b)-F_X(a)\]
- \(F_X(x)\) is non-decreasing \(x\)

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