I am trying to formulate an expression to calculate the critical value of a T-distribution for a given degrees of freedom. I have done so already for the Normal Distribution by considering the TI-84 invNorm function. According to this website: http://tibasicdev.wikidot.com/invnorm, the function can be expressed as such:$$invNorm(p,\mu,\sigma)=\mu+\sigma(\sqrt{2}\sum_{k=0}^\infty \frac{c_k}{2k+1}(\frac{\sqrt{\pi}}{2}(2p-1))^{2k+1}$$, where $$c_0=1$$ $$c_k=\sum_{m=0}^{k-1}\frac{c_mc_{k-1-m}}{(m+1)(2m+1)}$$The same website provides information for the invT function here: http://tibasicdev.wikidot.com/invt, but only says how the function could be expressed. It gives an explicit formula for 1df, but only mentions that the general case can be expressed in terms of the inverse incomplete beta function. For one, I am not able to find much information on an inverse incomplete beta function, so I am stuck determining the explicit formula. 2 other sources - https://www.mathworks.com/help/stats/tinv.html and http://www.statisticshowto.com/tables/inverse-t-distribution-table/ - provide formulas, but they don't seem to be what I am looking for as the latter mentions that the formula for P it gives is such that $$P=Pr(X\le x)$$which seems to behave more like a cumulative distribution function. I also tried evaluating the expression for values corresponding to a known critical value and the result wasn't even close. In all, is there a formula that I can use, like the one for z critical values above, to calculate the t critical value?


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