---
title: "Lab3"
author: "Zhou"
date: "2/3/2021"
output: html_document
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

## when sample size goes to infinity


```{r}
# Expectation = Mean when sample size goes to infinity.
par(mfrow=c(3,1),mar = c(3,4,1,1))
# The expectation of chi square here equals the df
x=rchisq(n=10,df=5)
hist(x)
abline(v=mean(x), col = "red")

x=rchisq(n=100,df=5)
hist(x)
abline(v=mean(x), col = "red")

x=rchisq(n=10000,df=5)
hist(x)
abline(v=mean(x), col = "red")

# Variance
## change the number size
n=100
x=rnorm(n,100,5)
hist(x)
c(min(x),max(x))
sum(x-mean(x))/(n-1)
sum((x-mean(x))^2)/(n-1)
sqrt(sum((x-mean(x))^2)/(n-1))

# var()
```

## Covariance

```{r}
# define 3 random variable
n=10000
a=rnorm(n,100,5)
x=a+rpois(n, 100)
y=a+rchisq(n,5)
z=a+rt(n,100)
par(mfrow=c(3,1),mar = c(3,4,1,1))
plot(x,y)
plot(x,z)
plot(y,z)

var(x)
var(y)
var(z)
cov(x,y)
cov(x,z)
cov(y,z)

W=cbind(x,y,z)
dim(W)
cov(W)
var(W)
cor(W)

```

## Matrix manipulations

```{r}
a=matrix(seq(10,60,10),2,3)
b=matrix(seq(1,6),2,3)
a
b
# add
a+b
# subtraction
a-b
# (dot)product
a*b
# (dot)division
a/b
# matrix multiplication
c=matrix(c(1,1,1,1,3,4,2,1,30,50,40,25),4,3)
b=matrix(c(20000,10000,1000,1000,300,20),3,2)
t=c%*%b
# inverse
t = t[1:2,1:2]
ti=solve(t)
ti
ti %*% t
t%*%ti
# transpose
c=matrix(c(1,1,1,4,30,50),2,3)
c
t(c)

# Properties of transpose
A=matrix(c(1,1,1,4,30,50),2,3)
B=matrix(c(1000,300,20,20000,10000,1000),3,2)
t(A%*%B)
t(B)%*%t(A)

```