# 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.

gsl_ran_tdist(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 $$-\infty < x < +\infty$$.

gsl_ran_tdist_pdf(x, 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.

gsl_cdf_tdist_P(x, nu)
gsl_cdf_tdist_Q(x, nu)
gsl_cdf_tdist_Pinv(P, nu)
gsl_cdf_tdist_Qinv(Q, nu)

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