4.3

The Chi-square Distribution

If $Z_1, Z_2,\cdots,Z_n$ are independent standard normal random variables, then $X=Z^2_1+Z^2_2+\cdots+Z^2_n$ follows a Chi-square distribution with $n$ degree of freedom, denoted by $X\sim \mathcal X^2_n$.

• Probabilities related to Chi-square distribution are obtained from Chi-square table or computer.
• If $X_1$ and $X_2$ are independent Chi-squared random variables where $X_1\sim \mathcal X^2_{n_1}$ and $X_2\sim \mathcal X^2_{n_2}$, then $X_1 + X_2 \sim \mathcal X^2_{n_1+n_2}$.

4.3.1 $\mathcal X^2$-distribution

An extension of Normal distribution $X\sim N(\mu,\sigma^2)$

• We now consider the sampling distribution of $\mathcal X^2= \frac{(n-1)S^2}{\sigma^2}$.
• Considering $\sum^n_{i=1}(X_i-\mu)^2$,

Theorem

If $S^2$(sample variance, $S^2=\frac{\sum(x_i-\bar x)^2}{n-1}$) is the variance of a random sample of size $n$ taken from a normal population having variance $\sigma^2$, then the statistic $\mathcal X^2\sim \frac{(n-1)S^2}{\sigma^2}$ has a chi-square distribution with $v=n-1$ degrees of freedom.

4.3.2 How does the $\mathcal X^2$-Distribution look like

The most important thing is: The $\mathcal X^2$ outside the value between $\mathcal X^2_{0.975}$ and $\mathcal X^2_{0.025}$ is not likely to occur unless our assumed value of $\sigma^2$ is too small. In a word, used for confirm whether an assumption is true. (see example and 4.3.3)

Chi-Square Distribution Table

Example

4.3.3 What is the $\mathcal X^2$-Distribution Used For?

The $\mathcal X^2$-Distribution is commonly used for statistical inferences.

Understand with the example above.