---
title: "Distributions&inference"
author: "Zhou"
date: "1/27/2021"
output: html_document
---

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


## Binomial in R
First, we will use the "rbinom" function to generate random numbers which belong to the Binomial distribution.
```{r}
p=.5
n=5 #number of layers/trials
k=10000 #number of balls
x=rbinom(k, n, p)
hist(x)

```

Second, we will try different probability and trials.

```{r}
p=.4
n=200 #number of layers/trials
k=10000 #number of balls
x=rbinom(k, n, p)
hist(x)
```

Third, standardization for the vector x_2.

```{r}
mean=n*p
var=n*p*(1-p)
z=(x-mean)/sqrt(var)
hist(z)
```

Fourth, let's plot on the probability density

```{r}
d=density(z)
par(mfcol=c(2,1),mar = c(3,4,1,1)) # define the graphical parameters (canvas) 
plot(d)
plot(d)
polygon(d, col="red", border="blue")
# dev.off() # close the canvas
```

## Normal distribution in R

Define the normal distribution and plot the probability density of the normal distribution.

```{r}
# Normal distribution
mean=n*p   # k = 10000, n=5, p=0.5
var=n*p*(1-p)
x=rnorm(k, mean=mean,sd=sqrt(var))
hist(x)
d=density(x)
plot(d)
```

Second, add two normal distribution variables

```{r}
par(mfrow=c(1,3),mar = c(3,4,1,1))
k=10000
x1=rnorm(k,0,1)
x2=rnorm(k,0,1)
y=x1^2 + x2^2   # you create a new distribution derived from 2 normal distribution
mean(y)
var(y)
hist(x1)
hist(x2)
hist(y)
```


## Poisson, Chi square, F and t distribution in R

```{r}
par(mfrow=c(1,4),mar = c(3,4,1,1))
lambda=.5
x_Poisson=rpois(k, lambda)
hist(x_Poisson)

x_Chi=rchisq(k,2)
hist(x_Chi)

x_F=rf(k,1, 100)
hist(x_F)

x_t=rt(k,2)
hist(x_t)
```

## Central Limit Theory (CLT)

Averages of large samples close to normal distribution.

First, let's see how to sample from different distributions.

```{r} 
par(mfrow=c(5,1),mar = c(3,4,1,1))
#Binomia
p=.05
n=100 #number of layers/trials
k=10000 #number of balls
x=rbinom(k, n,p)
d=density(x)
plot(d,main="Binomial")
#Poisson
lambda=10
x=rpois(k, lambda)
d=density(x)
plot(d,main="Poisson")
#Chi-Square
x=rchisq(k,5)
d=density(x)
plot(d,main="Chi-square")
#F
x=rf(k,10, 10000)
d=density(x)
plot(d,main="F dist")
#t
x=rt(k,5)
d=density(x)
plot(d,main="t dist")

```

Second, write a Function to get mean of ten. These ten is sampled from a distribution. Then plot the mean.

```{r}
i2mean = function(x,n=10){
  k=length(x)
  nobs=k/n
  # get matrix: nobs (1000) rows and n (10) cols  
  xm=matrix(x,nobs,n)
  # get means of each row
  y=rowMeans(xm)
  # we can get 1000 means
  return (y)
}

par(mfrow=c(5,1),mar = c(3,4,1,1))
#Binomia
p=.05
n=100 #number of layers/trials
k=10000 #number of balls
x=i2mean(rbinom(k, n,p))
d=density(x)  # get the density of mean
plot(d,main="Binomial")
#Poisson
lambda=10
x=i2mean(rpois(k, lambda))
d=density(x)  # get the density of mean
plot(d,main="Poisson")
#Chi-Square
x=i2mean(rchisq(k,5))
d=density(x)  # get the density of mean
plot(d,main="Chi-square")
#F
x=i2mean(rf(k,10, 10000))
d=density(x)  # get the density of mean
plot(d,main="F dist")
#t
x=i2mean(rt(k,5))
d=density(x)  # get the density of mean
plot(d,main="t dist")

```