# t - Distribution (Student t - Distribution)

t-Distribution is also known as the Student t-Distribution. If Z has a standard normal distribution N(0;1) and Y, the sum of $df$ squared standard normal random variables, has a $\chi ^{2}$ -distribution with $df$ degrees of freedom , then we define $T={\frac {Z}{\sqrt {\frac {Y}{df}}}}$ as the t-distribution with parameter $df$ (shortly written as t($df$ )), if Z and Y are independent. The parameter $df$ represents the degrees of freedom for the $\chi ^{2}$ random variable Y. The random variable T has range $-\infty \leq T\leq +\infty$ and expected value and variance: $E(T)=0,\ {\text{for }}df>1$ $Var(T)=f/(f-2),\ {\text{for }}df>2$ The following diagram plots the density function a t-distribution for different numbers of degrees of freedom $df$ .
. The Chi-square, t-, and F- distributions are distributions that are functions of Normal random variables that are particularly useful in statistics. On the t-Distribution. The density function of a t-distribution is a bell-shaped symmetricdistribution with expected value $E(T)=0$ (as a standard Normal distribution). However, a t-distribution has heavier tails than a standard Normal distribution. In other words, the t-distribution will be more dispersed than a standard Normal distribution. The variance of the standard normal distribution is 1, but the variance of a t-distribution equals $Var(T)=df/(df-2)$ (for $df>2$ ). As $df\rightarrow \infty$ , the density function of the t-distribution converges to the standard normal distribution. For $df\geq 30$ , a Normal distribution can produce a good approximation to a t-distribution. The t-distribution is tabulated different values of $df$ .