Formula or procedure for computing standard statistical tables such as z table, Student's t-table, or chi-square table How can the following tables be computed:


*

*Chi-Square table

*Student t-table


I'm looking for a formula or procedure used to make these tables.
For example:
If I have a given 'x'-value, as df(degree of freedom) with some confidence percentage 'y', I should be able to plug these x and y values into that formulae or procedure and should get results close to that in the standard tables.
I'm pretty sure that there is some logic behind these tables, but I could not find any concrete pointers. how to reach them. My guess, about some logic got surety when I found some tool doing this online: http://faculty.vassar.edu/lowry/csqsamp.html
 A: On e.g. wikipedia you can find the formula for the cdf / pdf of these distributions. Enter the values of the parameters and you're done.
If you want the reverse (I think you do from your question), simplest 'general' way of getting it done (not all cdfs have an analytic inverse) is use a univariate solver. 
Maybe you could just use R, that holds all these functions...
A: I would look into a few special functions, which continually "pop-up" in statistics - often in disguised forms:


*

*the confluent hypergeometric function $_{1}F_{1}(a;b;z)$.  heaps of functions are special cases of this one, such as erf, incomplete gamma, bessel.

*the incomplete gamma function (and regularised incomplete gamma function)

*the incomplete beta function (and regularised incomplete beta function)

*Gaussian hypergeometric function $_{2}F_{1}(a;b;c;z)$


2) and 3) are the CDFs for chi-square and t-distributions for certain values of the parameters.  I would invert these using the table backwards, rather than directly looking for an inverse function.  It is likely that the error in interpolation (for sufficiently close values) may be less than the error in numerically evaluating a complicated inverse function.
