Chi-Square distribution

Chi-Square distribution 

Chi-Square distribution is a distribution with parameter n which is called 'degree of freedom.'

The distribution is closely related to Normal distribution and Gamma distribution. 

Specifically, it is a critical distribution when it comes to samples and variance of normal distribution. 

Also, it is a special case of Gamma distribution with specific parameters. 

<PDF, expected value and variance of Chi-square distribution>

<The way to interpret Chi-square table>

<Moment generating function of chi-square distribution>

<Property1 - Sum of independent and identical Chi-square distribution>

The first property of Chi-square distribution is that sum of independently and identically distributed chi-square distribution becomes also Chi-distribution. We can prove It using Moment generating function of Chi-square distribution. 

<Property1> Central limit theorem of Chi-square distribution>

With property1, you can express Chi-square distribution, n degrees of freedom is the sum of n independent and identically distributed Chi-square distribution with degree of freedom 1. 

By this property, Chi-square distribution follows Central limit theorem (the sum of independent and identical distributions tends toward Normal distribution as n gets larger.)

<Property2- Relationship Chi-square distribution and Standard Normal distribution>

Squared Standard Nomral distribution equals to Chi - square distribution with 1 degree of freedom. You can prove it with moment generating function of squared standard normal distribution. 

According to the first property of Chi-square distribution, you can express Chi-square distribution with n degrees of freedom as the sum of n independent and identically distributed Chi-square distribution with 1 degree of freedom.

So, Chi-square distribution with n degrees of freedom equals n independent squared standard normal distribution.

n degrees of freedom and n-1 degrees of freedom is definitely different. chi-square distribution with n degrees of freedom is related to the total mean and chi-square distribution with n-1 degrees of freedom is the distribution related to sample mean. You should distinguish n degrees of freedom from n-1 degrees of freedom. 

<Property3- Relationship Chi-square distribution and Gamma distribution>

Chi-square distribution with n degrees of freedom is same with Gamma distribution when parameter of Gamma , k equals n/2, θ equals 2. 

<Property4- Relationship Chi-square distribution and Gamma distribution>

The relationship between gamma distribution and Chi-square distribution and the relationship between exponential and gamma distribution makes the relationship between exponential and Chi-square distribution. 

<Property5- Variance and Sample variance of Normal distribution >

This is the most important property of Chi-square distribution. 

When there are n samples from normal distribution, (n-1)*sample variance / variance becomes to follow Chi-square distribution with n-1 degrees of freedom. 

<Chi-square plot>

<Example >

<Plot and Calculation with R>

> pchisq(5.6,df=9)

[1] 0.2208123

> 1-pchisq(5.6,df=9)

[1] 0.7791877

> qchisq(1-0.77918,df=9)

[1] 5.60008

> curve(dchisq(x,df=9),from=0, to=30)

> x<-seq(5.6,30)
> plot<-dchisq(x,df=9)
> polygon(c(x, rev(x)), c(plot, rep(0, length(plot))))
> polygon(c(x, rev(x)), c(plot, rep(0, length(plot))),col = adjustcolor('red', alpha=0.3),border=NA)