### 20.20 The t-distribution

The t-distribution arises in statistics. If Y_1 has a normal distribution and Y_2 has a chi-squared distribution with \nu degrees of freedom then the ratio,

    X = { Y_1 \over \sqrt{Y_2 / \nu} }

has a t-distribution t(x;\nu) with \nu degrees of freedom.

Function: double gsl_ran_tdist (const gsl_rng * r, double nu)

  This function returns a random variate from the t-distribution. The distribution function is,

      p(x) dx = {\Gamma((\nu + 1)/2) \over \sqrt{\pi \nu} \Gamma(\nu/2)}
         (1 + x^2/\nu)^{-(\nu + 1)/2} dx

  for -\infty < x < +\infty.

Function: double gsl_ran_tdist_pdf (double x, double nu)

  This function computes the probability density p(x) at x for a t-distribution with nu degrees of freedom, using the formula given above.

Function: double gsl_cdf_tdist_P (double x, double nu)
Function: double gsl_cdf_tdist_Q (double x, double nu)
Function: double gsl_cdf_tdist_Pinv (double P, double nu)
Function: double gsl_cdf_tdist_Qinv (double Q, double nu)

  These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the t-distribution with nu degrees of freedom.