Distribution of Process capability (Cp) is given by the below formula. $$Cp\sqrt{\frac{\chi_\nu^2}{\nu}}$$ $\nu$ = The degrees of freedom
To calculate mean and variance of $\sqrt{\frac{\chi_\nu^2}{\nu}}$ distribution, I referred to the below link. What's the expectation of square root of Chi-square variable?.
And ChatGPT returns the below formula.
The mean of X is: $$E[X] = \frac{\sqrt{2}}{\sqrt{\nu}}\frac{\Gamma(\frac{\nu+1}{2})}{\Gamma(\frac{\nu}{2})}$$
The variance of X is: $$Var(X)=1-\frac{2}{\nu}\left(\frac{\Gamma(\frac{\nu+1}{2})}{\Gamma(\frac{\nu}{2})}\right)^2$$
But my calculation result by the above formula is different from the random data generation result. (I created random data to meet the target Cp and calculate the mean)
For example, in case of $\nu=9$, mean by the above formula is 0.972659274
But mean by random data generation (1,000,000 times creation) is 1.094479469
Considering the asymmetric Chi_squared distribution, mean of $\sqrt{\frac{\chi_\nu^2}{\nu}}$ should be larger than 1. And it should be converged to 1 by the larger $\nu$ value.
Would you please advice how to calculate the mean and variance for square root of chi-squared distribution divided by the degrees of freedom ?
