T distribution

T distribution

T distribution is a distribution with a degree of freedom which is usually notated as n. 

The distribution is very important distribution because it is frequently used for statistical analysis such as t-test. 

Also the distribution is highly related to distributions such as F distribution, Normal distribution and Chi-square distribution. 

<pdf, expected value, variance of T distribution>

As you can see through the graph, T distribution is symmetric by x=0 like standard normal distribution.

And as n goes to infinity, the variance goes to 1 and it converges to standard normal distribution which has 0 as mean and 1 as variance. 

<The way to interpret the graph of F distribution >

<T distribution with degrees of freedom n >

T distribution with degrees of freedom n is derived by chi square distribution with degrees of freedom n and standard normal distribution. 

<T distribution with degrees of freedom n-1 >

When we consider the sample mean and sample variance of random samples from Normal distribution, we can find the relationship between the sample mean and variance with t distribution.  

As I mentioned above, degrees of freedom n of T distribution is derived from Chi-square distribution with degrees of freedom n. 

That is, degrees of freedom of T distribution becomes n-1 when it is related to Chi-square distribution with n-1. As you know, Chis square distribution with degrees of freedom n-1 is the distribution of (n-1) times sample variance over population variance. we can have the form of t distribution with degrees of freedom n-1. 

< Relation to F distribution >

< Convergence to Standard Normal distribution >

as n goes to infinity, T distribution converges to N(0,1). As sample variance converges to population variance. 

> x<-seq(-5,5,0.01)

> df_1<-dt(x,1)

> df_5<-dt(x,5)

> df_20<-dt(x,20)

> T<-data.frame(x,df_1,df_5,df_20)

> 

ggplot(data=T,aes(x))+geom_line(aes(y=df_1,col="df_1"))+geom_line(aes(y=df_5,col="df_5"))+geom_line(aes(y=df_20,col="df_20"))+geom_line(aes(y=dnorm(x,0,1),col="norm"))+labs(y="density",title="T distribution")+scale_colour_manual(values=c("df_1"="red","df_5"="blue","df_20"="yellow","norm"="black"))

< A property of T distribution>