# Hello World! 🌎

Hello internet, I am a new user 🐣! Just trying out the medium platform, let's see how it goes! And to include some tips for me for the future 👍.

# Display some of my favorite Math:

**Einstein’s famous Mass-energy equivalence formula 🌌:**

**From wiki:** The formula defines the energy *E** *of a particle in its rest frame as the product of mass ** m** with the speed of light squared (

**).**

*c²***The 🔔 Curve :**

**With the probability density function:**

Where ** μ** is the mean or expectation of the distribution and

*σ**is the standard deviation. i.e. variance of*

*σ²**.*It is also known as the

*Gaussian Distribution*or more commonly known as the

*Normal Distribution*.

**An example graph of the function:**

To begin a code block in the medium editor:

**Windows/Linux:**Ctrl + Alt + 6**Mac:**Command + Option + 6- or type ``` (triple backtick) on a new line

#Clear console and environment

rm(list = ls())

cat(“\014”)#attach required library

library(ggplot2)

library(dplyr)#set mu and sigma parameters for the Normal Curve

mu<-0

sigma<-1#generate data

x<-seq(-4*sigma,4*sigma,0.0001)-0.5

y<-dnorm(x,mu,sigma) #density functiondata<-cbind(x=x,y=y) %>% data.frame()

xlab<-c(expression(-3*sigma)

,expression(-2*sigma)

,expression(-sigma)

,expression(mu)

,expression(sigma)

,expression(2*sigma)

,expression(3*sigma)

)p_lab<-pnorm(seq(-3*sigma,3*sigma,sigma),mu,sigma)#Plot:

ggplot(data,aes(x,y))+

#Plot area settings:

theme_classic()+

theme(plot.title=element_text(size=40,face=”bold”,hjust=0.5)

,panel.grid.major.x=element_line(size = (0.2))

,axis.text.x=element_blank())+

ggtitle(“Standard Normal Distribution”)+

xlab(“x”)+

#text for the sigma labels

annotate(“text”,

x = seq(-3*sigma,3*sigma,sigma),

y = rep(-0.005,7),

label = xlab,

family = “”, fontface = 3, size=8) +

#text for the density values at each n*sigma area

annotate(“text”,

x = c(seq(-3*sigma,3*sigma,sigma)-0.5*sigma,0.5*sigma+3*sigma),

y = round(c(p_lab[1],diff(p_lab),p_lab[1]),1)²+0.05,

label = paste0(round(c(p_lab[1],diff(p_lab),p_lab[1])*100,1),”%”),

family = “”, fontface = 8, size=8) +

#display parameter values

annotate(“text”,

x = rep(2.2*sigma,2),

y = c(0.16,0.14),

label = c(expression(paste(mu,”=”)),

expression(paste(sigma,”=”))),

family = “”, fontface = 3, size=8) +

annotate(“text”,

x = rep(2.4*sigma,2),

y = c(0.164,0.143),

label = c(mu,sigma),

family = “”, fontface = 3, size=7) + scale_x_continuous(breaks=seq(-3*sigma,3*sigma,sigma),limits=c(-3.5*sigma,3.5*sigma))+

ylab(“Probability Density”)+

scale_y_continuous(labels=scales::percent_format(accuracy=1))+

geom_line()

**Maybe some Linear Algebra:**

To find a Plane of best-fit.

Given a data vector with** n** samples and

**parameters:**

*p*Where the dependent variable *y *and the ** p**-size vector of regressors

*x**are assumed to be a linear relationship. Where the error variable*

**ε**was modeled such that it is the minimum and ideally it experiences an unobserved random variable. i.e. “

*noise*”.

Then ideally we want to find *β *where the model has the form:

Or simply

Where

and

**Least-squares estimation (a.k.a. Line of best fit):**

Since *y* and *x *are assumed to be a linear relationship and we would like to find the “best” ** β** which solve the system of equations and to minimize

**. Hence let:**

*ε*** L** is called the

*Loss*function, essentially the error term is modeled with the input

*X, y, β**.*Since

**is the original data we want to “fit”, therefore we can find the “best”**

*X, y*

*β**at which*

*L(X, y, β)**is minimized.*

Hence the first derivative of ** L(X, y, β)**:

Since ** X, y** is fixed and known and we only interested in the “best”

**therefore:**

*β*This is the case for the Simple Linear Regression.

This concludes the Hello World! 🌎

Thank you for reading!