A Noob’s perspective — Learning Data Science, ML and AI from scratch. Pt 2

Statistics fundamentals

Before even getting into statistics and probability, i’ll make some refresher notes of different essential topics like, set theory and calculus

Set theory

Definition

• a set is a collection or group of items. those items are usually called elements

Notation

• Upper case letters usually for sets, A, B, C, …
• Lower case letters for elements a, b, c, …
• ∈ to denote membership and ∉ to denote the opposite
• U is the universal set
• ∅ is a null set
• Some examples
• A = {1,2,3, …, 10}, 2 ∈ A; 33 ∉ A
• B = {x| 0 <= x <= 1} “|” means such that so b is the set that contains all the numbers greater or equal to 0 and less or equal to 1.

more definitions

• If every element of A is contained within B, then A is a subset of B, it’s denoted as A ⊆ B
• A is equal to B if and only if A ⊆ B and B ⊆ A

Properties

• ∅ ⊆ of any set, the null set is a subset of any set
• Any set is ⊆ U, any set is a subset of the universal set
• A ⊆ A, any set is a subset of itself
• If A ⊆ B and B ⊆ C then A is also ⊆ of C

Even more definitions

• The complement of A or A is = {z| x ∈ U and x ∉ A} or any element of the universe that is not contained in A.
• The intersection of A and B is A ∩ B {x| x ∈ A and x ∉ B}
• The union is A U B {x| x ∈ A or x ∉ B}
• Minus is B — A = B ∩ A

Symmetric difference or XOR

• A Δ B = (A — B) U (B — A) = (A U B) — ( A ∩ B)
• or simply put everything in the interception except what’s left of A and B

Cardinality of a set is denoted, for example as: is A is a set the cardinality is |A|, and refers to the amount of elements that a set has.

Example

• A = {3,4} is finite, since |A| = 2.
• B = {1,2,3, ….} is countably infinite. i.e |B| = No
• C = {x| x ∈ [0,1]} is uncountably infinite. i.e |C| = N1

Laws of Operations

• Complement Law: A U Ā = U, A ∩ Ā = Ø, Ā = A
• Commutative: A U B = B U A , A ∩ B = B ∩ A
• Associative: A U (B U C) = (A U B) U C, A ∩ (B ∩ C) = (A ∩ B) ∩ C
• Distributive: A U (B ∩ C) = (A U B) ∩ (A U C) , A ∩ (B U C) = (A ∩ B) U (A ∩ C)
• DeMorgan’s:

next topic is calculus.