PROBABILITY & STATISTICS

Probability

measure of one’s belief in the possible occurence of an event

Random/Stochastic Events

• cannot be predicted with certainty

• have stable relative frequencies over long period of trials

Relative Frequency

The number of p occurences observed in n trials, divided by n, when n > 0.

• as n increases, the limit of this may approach the value of the probability, making it merely an estimate

Population

set of values that can be generated as the scenario is performed ad infinitum

Hypothesis

an inferred statement made to test a point; may seek to contradict this using observation/experimentation

Highly Improbable

a result that is very unlikely though not impossible; eg. an astronomically low p-value

Set

a collection of distinct objects/values that share a common property

Union

the elements in either or both of two sets

Intersection

the set of values two sets share in common

Compliment

the set of points outside of a specific set

Pairwise Disjoint

2 sets that are mutually exclusive; share nothing in common.

Distributive Laws

Intersections distribute over unions.

• A (intersects) (B U C) <=> (intersection of A & B) U (intersection of A & C)

De Morgan’s Laws

The compliment of intersections is the union of compliments (or vice versa: the compliment of unions is the intersection of compliments)

• (A U B)^c = intersection of A^c & B^c

Experiment

A process for which an observation is made

Event

An outcome of an experiment

Simple event

An event that cannot be broken down any further

• has unique sample points

• are disjoint to each other if an experiment is only performed once

Compound event

An event that CAN be broken down further into more specific events; “broad event“

Sample point

BIJECTIVE (one-to-one and onto) with a sample event

Sample space

The set of all possible points

Countable

True when there exists a set that has a bijection with itself + natural numbers

Discrete sample space

Exists if the cardinality is either finite or countable

Discrete event

any subset of discrete sample space

• in other words: collection of sample points

Discrete probabilistic model

Assigns numerical probabilities to each simple event found in the discrete sample space S

Sample-point method

Finds probability of an event A in sample space S, when S is at most countable.

Steps of sample-point method

1) DEFINE

• sample space - by listing sample events

• simple events

• experiments

2) ASSIGN PROBABILITIES

• all probabilities must sum to 1 and be less than or equal to 0 separately

3) DEFINE A

• the union of all applicable simple events

• test each point

4) FIND P(A)

• sum up all the simple events in A to get the probability