2.2 Chi-squared distribution
Definition 2.8 noncentral Chi-squared distribution
- $k \in \mathbb{N}$,
- $\mu \in \mathbb{R}$,
- $X_{1} \sim N(\mu, 1)$,
- $X_{j} \sim N(0, 1) \ (j = 2, \ldots, k)$,
- $X_{1}, \ldots, X_{k}$,
- independent
The distribution of r.v. $Y := \sum_{j=1}^{k} X_{j}$ is said to be chi-suqre distribution with $k$ degree of freedom and noncentrality parameter $\mu^{2}$. The p.d.f of $Y$ is given by
\[f(x) = e^{-\frac{\mu^{2}}{2}} \sum_{r=0}^{\infty} \frac{1}{r!} \left( \frac{\mu^{2}}{2} \right)^{r} g(x; \frac{1}{2}, r + \frac{k}{2}) \quad (x > 0)\]where $g(x; \alpha, \nu)$ is the p.d.f. of gamma distribution with $\alpha, \nu$. We denote $\chi^{2}(k ,\mu^{2})$ by chi-suqre distribution with $k$ degree of freedom and noncentrality parameter $\mu^{2}$. In particular, chi-squared distribution with $k$ degree of freedom and denote $\chi^{2}(k) := \chi^{2}(k, \mu^{2})$ when $\mu = 0$. In this case, the p.d.f. of $\chi^{2}(k)$ is given by
\[f(x) = \sum_{r=0}^{\infty} \frac{1}{r!} \left( \frac{1}{2} \right)^{2} g(x; \frac{1}{2}, r + \frac{k}{2}) \quad (x > 0)\]