20.4.8  Student’s distribution

The probability density function for Student’s distribution.

Student’s distribution (also called Student’s t-distribution or the t-distribution) with n degrees of freedom has density function given by

where recall the Gamma function (see Section 7.3.13) is defined for x>0 by

The student or studentd command finds the density function for Student’s distribution.

• student takes two arguments:
  • n, an integer (the degrees of freedom).
  • x, a real number.
  student(n,x) returns the value of the density function for Student’s distribution with n degrees of freedom at x, given in (5).

Example

which can be numerically approximated by:

The cumulative distribution function for Student’s distribution.

The student_cdf command computes the cumulative distribution function for Student’s distribution.

• student_cdf takes two mandatory arguments and one optional argument.
  • n, an integer (the degrees of freedom).
  • x, a real number.
  • Optionally, y, a real number.
• student_cdf(n,x) returns Prob(X ≤ x) for Student’s distribution with n degrees of freedom.
• student_cdf(n,x,y) returns Prob(x ≤ X ≤ y).

Examples

The inverse distribution function for Student’s distribution.

The student_icdf command computes the inverse distribution for Student’s distribution.

• student_icdf takes two arguments:
  • n, an integer (the degrees of freedom).
  • h, a real number between 0 and 1.
• student_icdf(n, h) returns the inverse distribution for Student’s distribution with n degrees of freedom; namely, the value of x for which Prob(X ≤ x)=h.

Example

The upper tail cumulative function for Student’s distribution.

The UTPT (the Upper Tail Probability-T distribution) computes Prob(X > x) for Student’s distribution.

• UTPT takes two arguments:
  • n, an integer (the degrees of freedom).
  • x, a real number.
• UTPT(n,x) returns Prob(X > x) for Student’s distribution with n degrees of freedom.

Example