Non-central chi-squared distribution
Find the probability density function, lower cumulative distribution function, and upper cumulative distribution function of the non-central chi-squared distribution.
・If λ = 0, this corresponds to the chi-squared distribution.
| \(\normalsize Noncentral\ Chi-Squared\ distribution\\ \hspace{220px} X^2(x,\nu,\lambda)\\ (1) probability\ density\\ \hspace{30px}f(x,\nu,\lambda)={\large\displaystyle \sum_{\small j=0}^{\small \infty}\frac{e^{-\frac{\lambda}{2}}(\frac{\lambda}{2})^j}{j!}\frac{x^{\frac{\nu+2j}{2}-1}e^{-\frac{\small x}{2}}}{2^{\frac{\nu+2j}{2}}\Gamma(\frac{\nu+2j}{2})}}\\ (2) lower\ cumulative\ distribution\\ \hspace{30px}P(x,\nu,\lambda)={\large\int_{\small 0}^{\small x}}f(t,\nu,\lambda)dt\\ (3) upper\ cumulative\ distribution\\ \hspace{30px}Q(x,\nu,\lambda)={\large\int_{\small x}^{\small\infty}}f(t,\nu,\lambda)dt\\\) |
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