Calculating quantiles for chi squared distribution

Question

For chi squared distribution, how would I find quantiles for the following three cases:

A) $P(X^2> X^2_{\alpha})=0.01$ when $v = 21$

B) $P(X^2 < X^2_{\alpha})=0.95$ when $v =6$

C) $P(X^2_{\alpha} < X^2 <23.209) = 0.015$ when $v = 10$

Here $X^2_{\alpha}$ is the $\alpha$-quantile of the $\chi^2_{v}$ distribution.

Answer 1

By hand, you need to refer to a tabulated distribution of the $\chi^2$, which should be found easily on the web (e.g., this one). Let x denotes the quantile of interest, and v the degrees of freedom of the chi-square distribution. You just have to know that the total area under the curve (i.e., the density) equals 1 (this will help you to work through the third case), and that such Tables generally gives P(X < x) = p, for a certain v. Knowing x (resp. p), you can find the approximated value of p (resp. x). In the aforementioned Table, the first cell reads: P(X < 1.32) = 0.25, for a 1-df chi-square.

If you have R, the qchisq() function gives you the requested quantiles (look at the on-line help to be sure of what is returned, esp. the lower.tail argument). For the preceding example, we would use qchisq(0.25, 1, lower.tail=FALSE).

It is always a good idea to draw the corresponding density curve, as illustrated below. Note that p3 is also 1-P(X < q2).