# Probability Review

Updated 2017-10-20

## Axioms of Probability

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

1. Probability must be between 0 and 1: $$0\leq \mathbb P(A) \leq 1$$ for any event $$A$$
2. If events $$A$$ and $$B$$ are mutually exclusive ($$A \cap B=\emptyset$$) then $$\mathbb P(A\cup B)=\mathbb P(A) + \mathbb P(B)$$
3. 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:

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

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