Next: Noncentral Student's t Random Variable, Previous: Normal Random Variable, Up: Functions and Variables for continuous distributions [Contents][Index]
Student’s t random variable is denoted by t(n) where n is the degrees of freedom with n > 0. If Z is a \({\it Normal}(0, 1)\) variable and V is an independent \(\chi^2\) random variable with n degress of freedom, then
$$ Z \over \sqrt{V/n} $$has a Student’s t-distribution with n degrees of freedom.
Returns the value at x of the density function of a Student random variable t(n), with n>0 degrees of freedom. To make use of this function, write first load("distrib").
The pdf is $$ f(x; n) = \left[\sqrt{n} B\left({1\over 2}, {n\over 2}\right)\right]^{-1} \left(1+{x^2\over n}\right)^{\displaystyle -{n+1\over 2}} $$
Returns the value at x of the distribution function of a Student random variable t(n), with n>0 degrees of freedom.
The cdf is $$ F(x; n) = \cases{ 1-\displaystyle{1\over 2} I_t\left({n\over 2}, {1\over 2}\right) & $x \ge 0$ \cr \cr \displaystyle{1\over 2} I_t\left({n\over 2}, {1\over 2}\right) & $x < 0$ } $$
where \(t = n/(n+x^2)\) and \(I_t(a,b)\) is the beta_incomplete_regularized function.
(%i1) load ("distrib")$
(%i2) cdf_student_t(1/2, 7/3);
7 1 28
beta_incomplete_regularized(-, -, --)
6 2 31
(%o2) 1 - -------------------------------------
2
(%i3) float(%); (%o3) 0.6698450596140415
Returns the q-quantile of a Student random variable t(n), with n>0; in other words, this is the inverse of cdf_student_t. Argument q must be an element of [0,1]. To make use of this function, write first load("distrib").
Returns the mean of a Student random variable t(n), with n>0. To make use of this function, write first load("distrib").
The mean is $$ E[X] = 0 $$
Returns the variance of a Student random variable t(n), with n>2.
The variance is $$ V[X] = {n\over n-2} $$
(%i1) load ("distrib")$
(%i2) var_student_t(n);
n
(%o2) -----
n - 2
Returns the standard deviation of a Student random variable t(n), with n>2. To make use of this function, write first load("distrib").
The standard deviation is $$ D[X] = \sqrt{\displaystyle{n\over n-2}} $$
Returns the skewness coefficient of a Student random variable t(n), with n>3, which is always equal to 0. To make use of this function, write first load("distrib").
The skewness coefficient is $$ SK[X] = 0 $$
Returns the kurtosis coefficient of a Student random variable t(n), with n>4. To make use of this function, write first load("distrib").
The kurtosis coefficient is $$ KU[X] = {6\over n-4} $$
Returns a Student random variate t(n), with n>0. Calling random_student_t with a second argument m, a random sample of size m will be simulated.
The implemented algorithm is based on the fact that if Z is a normal random variable \({\it Normal}(0, 1)\) and S^2 is a \(\chi^2\) random variable with n degrees of freedom, \(\chi^2(n)\), then
$$ X={{Z}\over{\sqrt{{S^2}\over{n}}}} $$is a Student random variable with n degrees of freedom, t(n).
To make use of this function, write first load("distrib").