# Critical Values of the Chi-Square Distribution

I am trying to reproduce the (upper-tail) critical values of chi-square distribution with ν degrees of freedom as reported here. As a test I looked at $\nu=2$ and the probability $0.99$.

As far as I can tell, in Mathematica the ChiSquare distribution is implemented in ChiSquareDistribution, so here is what I tried to do to find the value:

NIntegrate[PDF[ChiSquareDistribution[2], x], {x, 99/100, Infinity}]

However, this gives 0.609571. Do I need to scale this value to get the desired result?

• Quantile[ChiSquareDistribution[2], .99] or NSolve[CDF[ChiSquareDistribution[2], t] == 99/100, t, Reals]?
– kglr
Dec 1, 2017 at 17:33
• InverseCDF[ChiSquareDistribution[2],99/100] as well.
– chuy
Dec 1, 2017 at 17:45
• Or working from the PDF: FindRoot[Integrate[PDF[ChiSquareDistribution[2], x], {x, 0, xf}] == 0.99, {xf, 1}] // Quiet or Assuming[xf > 0, Solve[Integrate[PDF[ChiSquareDistribution[2], x], {x, 0, xf}] == 0.99, xf]] // Quiet Dec 1, 2017 at 19:23
• For your question "Do I need to scale this value to get the desired result?", the answer is "No. You've formulated the statement incorrectly to obtain the desired result." To obtain 0.99 you'd need to use NIntegrate[PDF[ChiSquareDistribution[2], x], {x, 0, 9.21034}]. To obtain the desired value (9.21034) you'd need to follow any of the above comments.
– JimB
Dec 2, 2017 at 1:55

N @ Quantile[ChiSquareDistribution[2], 99/100]


9.21034

t /.  First@ NSolve[CDF[ChiSquareDistribution[2], t] == 99/100, t, Reals]


9.21034

Or, modifying the formulation in OP:

t /. First @ NSolve[Integrate[PDF[ChiSquareDistribution[2], x], {x, 0, t},
Assumptions -> Element[t, Reals]] == 99/100, t, Reals]


9.21034