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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").


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