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I would like to know the difference between the ##\chi^{2}## distribution and the PTE (Probability-To-Exceed) ?

I must compare 2 data sets A and B and in the article I am reading, they talk about this PTE :

For the moment, I only know the ##\chi^{2}## distribution with ##k=2## degrees of freedom :

##f(\Delta\chi^{2})=\dfrac{1}{2}\,e^{-\dfrac{\Delta\chi^{2}}{2}}\quad(1)##

and the relation with confidence level :

##1-CL={\large\int}_{\Delta\chi^{2}_{CL}}^{+\infty}\,\dfrac{1}{2}\,e^{-\dfrac{\Delta\chi^{2}}{2}}\,d\,\chi^{2}=e^{-\dfrac{\Delta\chi_{CL}^{2}}{2}}\quad(2)##

I don't know how to make the link with the image and text above. Indeed, in the article, they make appear the integral of gaussian whereas in ##(2)##, I can only make appear a simple integration of exponential (I mean, there is no "##\text{erf}##" function appearing unlike into the article).

If someone could explain the difference between ##\chi^{2}## distribution and ##P_{\chi^2}## PTE ?

Thanks

I must compare 2 data sets A and B and in the article I am reading, they talk about this PTE :

For the moment, I only know the ##\chi^{2}## distribution with ##k=2## degrees of freedom :

##f(\Delta\chi^{2})=\dfrac{1}{2}\,e^{-\dfrac{\Delta\chi^{2}}{2}}\quad(1)##

and the relation with confidence level :

##1-CL={\large\int}_{\Delta\chi^{2}_{CL}}^{+\infty}\,\dfrac{1}{2}\,e^{-\dfrac{\Delta\chi^{2}}{2}}\,d\,\chi^{2}=e^{-\dfrac{\Delta\chi_{CL}^{2}}{2}}\quad(2)##

I don't know how to make the link with the image and text above. Indeed, in the article, they make appear the integral of gaussian whereas in ##(2)##, I can only make appear a simple integration of exponential (I mean, there is no "##\text{erf}##" function appearing unlike into the article).

If someone could explain the difference between ##\chi^{2}## distribution and ##P_{\chi^2}## PTE ?

Thanks