4.4

4.4.1 Student $t$-Distribution

The standard deviation of the population, $\sigma$, is hard to obtain in reality, so we need a method to replace it with the sample standard deviation, $S$.

Theorem

Let $Z$ be a standard normal random variable ($Z\sim N(0,1)$), and $V$ be a chi-squared random variable ($V\sim \mathcal X^2_v)$. ($v$ is the df) If $Z$ and $V$ are independent, then the distribution of the random variable $T=\frac{Z}{\sqrt{\frac{V}{v}}}$ is given by the density function $h(t)=\frac{\Gamma(\frac{v+1}2)}{\Gamma(\frac{v}2)\sqrt{\pi v}}(1+\frac{t^2}{v})^{-\frac{v+1}2}$, $-\infin \lt t \lt \infin$.

This is known as the $t$-distribution with $v$ df(DOF).

Theorem - another

Given $X \sim N(0,1),Y\sim \mathcal X^2(n)$,

then $t = \frac{X}{\sqrt{\frac{Y}n}}$.

Corollary

• t-distribution table

4.4.2 How does the $t$-Distribution look like

• very similar to the standard normal distribution $Z$.
• variance depends on sample size $n$.
• As $n$ increases (meanwhile, $S\rightarrow\sigma$), $t$-Distribution becomes more likely to $Z$.

$t$-value $t_a$ is denoted as the point where the area above it equal to $a$.

• By symmetry about zero, $t_{1-a}=-t_a$

Example