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Phong Tran

Phong Tran

16 posts

Posts by Phong Tran

A reality example of Probability

Poker cards

52 cards at a deck with 4 type hearts, club, diamonds, spades, and including 13 for each.

  1. What is the probability of choose 2 red cards from this deck

52 cards: $\frac{26}{52} \cdot \frac{25}{51}$

  1. What is the probability of choose number 4 cards from this deck

52 cards, 4 cards having number 4: $\frac{4}{52}$

  1. What is the probability of choosing number 4 cards from this deck? Given these are black cards

52 cards, 4 cards having number 4 with 2 blacks and 2 reds: $\frac{2}{52}$

Numeric numbers

How many number

A reality example of Probability Read more

Infinite series

Definition

is an expression of the form:

$$ \sum_{n=1}^{\infty} a_n = a_1 + a_2 + \dots + a_i + \dots $$

The $N^{th}$partial sum $S_n$is the sum of the first N terms:

$$ S_n = a_1 + a_2 + \dots + a_N $$

We say the infinite series $\sum_{n=1}^{\infty} a_n$is convergent with sum $S$provided:

$$ S = \lim_{N \to \infty} S_N $$

if $\lim_{N \to \infty} S_N$does not exist, we say that the series diverges.

Examples

I'll provide an example of an infinite series and explain its convergence or divergence

Infinite series Read more

Notations at Math for ML

Notations at Math for ML
Photo by Saad Ahmad / Unsplash

This made from latex equations, so please go to website to see it phrase easier. Or try to download the extension called TeX for Gmail:

https://chromewebstore.google.com/detail/tex-for-gmail/gjnmclkoadjdljnfmbnnhaahilafoeji?hl=en

Basic Notation

Capital letters represent matrices: $A, B, C$
Lowercase letters represent vectors: $u, v, w$
Matrix $A$ has $m$ rows and $n$ columns: $A \in \mathbb{R}^{m \times n}$
The transpose of matrix $A$: $A^T$
The transpose of vector $v$: $v^T$
The inverse of matrix $A$: $A^{-1}$
The determinant of matrix $A$: $\det(A)$
Matrix multiplication of matrices $A$ and $B$

Notations at Math for ML Read more

Functions in math

Functions in math
Photo by Antoine Dautry / Unsplash

y = f(x), Domain: D

$$ f(x) = \sqrt{x+2}, D = x + 2 \ge 0 \leftrightarrow x \ge -2 $$

D(f) = [2, inf]

Polynomials

$$ p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 $$

For example

Type of Polynomials Degree Definition Example
Constant 0 simply a constant term. $f(x) = 5$  
Linear 1 represents a straight line $ f(x) = 2x - 3$  
Quadratic 2 represents a parabola $f(x) = x^2 + 4x + 1$
Cubic 3 $f(x) = 3x^3 - 2x^2 + x + 7$
Quartic 4 $f(x) = x^4 + 7x^3 -
Functions in math Read more

SETS, Notation

SETS, Notation
Photo by Roman Mager / Unsplash

In ordinary life

SET diagram in reality, made from Gemini and design into diagram by Lucidchart
SET diagram in reality, made from Gemini and design into diagram by Lucidchart

In mathematical

A intersection B (mutual value between A and B
A intersection B (mutual value between A and B
A Union B (takes all value from A and B
A Union B (takes all value from A and B

Notation

Notation Description Example
$x \in A$ The object $x$ is an element of the set $A$. If $A = \{1, 2, 3\}$, $2 \in A$.
$x \notin A$ The object $x$ is not an element of the set $A$. If $A = \{1, 2, 3\}$, $4 \notin A$.
$\emptyset$ The empty set, that is, the set with no elements at all. $\emptyset = \{\}$
$A \subseteq
SETS, Notation Read more

Propositional Logic, Truth tables, and Logical Equivalences

This made from latex equations, so please go to website to see it phrase easier. Or try to download the extension called TeX for Gmail:

Use truth tables to show which one is valid

P ∧ (Q ∨ R) ⇔ (P ∧ Q) ∨ (P ∧ R) (Distribution of ∧)

P Q R (Q ∨ R) (P ∧ Q) (P ∧ R) P ∧ (Q ∨ R) (P ∧ Q) ∨ (P ∧ R)
T T T T T T T T
T F F F F F F F
F T F T F F F F
F F T T F F F F
T F F F F F F F
F
Propositional Logic, Truth tables, and Logical Equivalences Read more