Student's t Random Variable (Maxima 5.47.0 Manual)

52.2.2 Student’s t Random Variable

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.

Function: pdf_student_t (x,n) ¶

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

\[f(x; n) = \left[\sqrt{n} B\left({1\over 2}, {n\over 2}\right)\right]^{-1} \left(1+{x^2\over n}\right)^{\displaystyle -{n+1\over 2}} \]

Categories: Package distrib ·

Function: cdf_student_t (x,n) ¶

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

Categories: Package distrib ·

Function: quantile_student_t (q,n) ¶: 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").

Categories: Package distrib ·

Function: mean_student_t (n) ¶

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

\[E[X] = 0 \]

Categories: Package distrib ·

Function: var_student_t (n) ¶

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

Categories: Package distrib ·

Function: std_student_t (n) ¶

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

\[D[X] = \sqrt{\displaystyle{n\over n-2}} \]

Categories: Package distrib ·

Function: skewness_student_t (n) ¶

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

\[SK[X] = 0 \]

Categories: Package distrib ·

Function: kurtosis_student_t (n) ¶

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

\[KU[X] = {6\over n-4} \]

Categories: Package distrib ·

Function: random_student_t (n)
random_student_t (n,m) ¶

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").

Categories: Package distrib · Random numbers ·