The chi-square distribution

Summary

• If $Z \sim N(0,1)$ then we say that $Y=Z^2$ follows a chi-square distribution with one degree of freedom and write

$$Y \sim χ_1^2$$

• If $Z_1,Z_2,…,Z_k$ are $n$ independent random variables and $Z_j \sim N\left( 0,1 \right), j=1,…,k$ then we say that $Y=Z_1^2+Z_2^2+…+Z_k^2$ follows a chi-square distribution with $k$ degree of freedom and write

$$Y \sim χ_k^2$$