MA2102

Probability and Statistics

Lecture-24

Student-t Distribution:

Let $X,Y$ are two independent random variables, $X\sim N(0,1)$, & $Y\sim \chi^2_n$, then $T=\frac{X}{\sqrt{\frac{Y}{n}}}=\frac{\sqrt{n}X}{\sqrt{Y}}$ is said to have a student-t distribution on $n-$degrees of freedom.

$f_X(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}$,$-\infty<x<\infty$, $f_Y(y)=\frac{1}{2^{\frac{n}{2}}\Gamma{\left(\frac{n}{2}\right)}}y^{\frac{n}{2}-1}e^{-\frac{y}{2}}$, $y>0$

The $JPDF$ of $X,Y$, $f_{X,Y}(x,y)=f_X(x)f_Y(y)$     ($\because$ $X,Y$ are independent)