# 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 ${\displaystyle df}$ squared standard normal random variables, has a ${\displaystyle \chi ^{2}}$-distribution with ${\displaystyle df}$ degrees of freedom , then we define ${\displaystyle T={\frac {Z}{\sqrt {\frac {Y}{df}}}}}$ as the t-distribution with parameter ${\displaystyle df}$ (shortly written as t(${\displaystyle df}$)), if Z and Y are independent. The parameter ${\displaystyle df}$ represents the degrees of freedom for the ${\displaystyle \chi ^{2}}$ random variable Y. The random variable T has range ${\displaystyle -\infty \leq T\leq +\infty }$ and expected value and variance: ${\displaystyle E(T)=0,\ {\text{for }}df>1}$ ${\displaystyle Var(T)=f/(f-2),\ {\text{for }}df>2}$ The following diagram plots the density function a t-distribution for different numbers of degrees of freedom ${\displaystyle df}$.
. 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 ${\displaystyle E(T)=0}$ (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 ${\displaystyle Var(T)=df/(df-2)}$ (for ${\displaystyle df>2}$). As ${\displaystyle df\rightarrow \infty }$, the density function of the t-distribution converges to the standard normal distribution. For ${\displaystyle df\geq 30}$, a Normal distribution can produce a good approximation to a t-distribution. The t-distribution is tabulated different values of ${\displaystyle df}$.