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

\[Y \sim χ_k^2\]