![[chi-square.png|300]] $ \begin{align*} \sum_{i = 1}^{k}Z^2_i &\sim \chi^2(k)\\ \chi^2(k) &= \frac{x^{\frac{k}{2} - 1}e^{-\frac{x}{2}}}{2^{\frac{k}{2}}\Gamma\paren{\frac{k}{2}}} \\ P(X \ge \chi^2_\alpha(k)) &= \alpha \end{align*} $ The Chi-Square ($\chi^2$) distribution is a family of [[Probability Distribution Function|PDFs]] that represents the sum of the squares of $k$ [[Probabilistic Independence|independent]], standard [[Normal Distribution|normal]] distributions, where parameter $k$ is the degrees of freedom. The distribution lies entirely to the right of the $y$-axis, and is skewed to the right for small values of $k$. Similar to the [[Poisson Distribution|Poisson distribution]], it approaches a normal distribution for larger values of $k$. It is used for [[Hypothesis Test|testing]] the [[Chi-Square Test|goodness of fit]] between a sample [[Frequency Distribution|frequency distribution]] and a theoretical distribution.