# prob-stat

Notes for reviewing my probability and statistics course atNortheastern.

# Sample Spaces

Experiment
Repeatable procedure with a set of possible results

Sample Outcome
One of many possible results of an experiment

Sample Space
$S$: The set of all possible outcomes of an experiment

Event
$A \subset S$ : 0 or more outcomes of an experiment

Probability
$P(A) = \frac{n(A)}{n(S)}$

# Set Theory

Discrete Set
A finite or countable set. i.e. tossing a coin until a head is received:$S = \{ H, TH, TTH, \dots \}$

Continuous Set
A range of values. I.e. getting a number smaller than 1 from within areal number between 0 and $\sqrt{2}$: $S =$, $A = [0, 1)$

Intersection
$A \cap B = \{x | x \in A\ and\ x \in B\}$ for some events A, B

Disjoint
$A \cap B = \emptyset$. Also known as "mutually exclusive"

Union
$A \cup B = \{x | x \in A\ or\ x \in B\}$

Complement
$A^{c} = \{x \in S | x \not{\in} A\}$

## DeMorgan's Law(s)

1. $(A \cup B)^{c} = A^{c} \cap B^{c}$
2. $(A \cap B)^{c} = A^{c} \cup B^{c}$

# Probability Function

$P$ assigns a real number to any event of a sample space, and followsthe following axioms for a sample space that's finite:

• $P(A) \geq 0$ for all $A$
• $P(S) = 1$
• $A \cap B = \emptyset \implies P(A \cup B) = P(A) + P(B)$, or$P(A \cup B) = P(A) + P(B) - P(A \cap B)$ otherwise
• $P(\cup_{i=1}^{\infty} A_{i}) = \sum_{i=1}^{\infty} P(A_{i})$ if any$A_1, A_2, A_3 \dots$ are mutually exclusive in $S$