9.4.8 Student’s distribution
The probability density function for Student’s distribution: student studentd
Student’s distribution (also called Student’s t-distribution or just
the t-distribution) with n degrees of freedom has density function
given by
student(n,x) =
| | ⎛
⎜
⎜
⎝ | 1 + | | ⎞
⎟
⎟
⎠ | |
(5) |
where recall the Gamma function (see Section 6.8.13) is defined for x>0 by
The student command finds the density function for Student’s
distribution.
studentd is a synonym for student.
-
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.
Input:
student(2,3)
Output:
which can be numerically approximated by:
Input:
evalf(student(2,3))
Output:
The cumulative distribution function for Student’s distribution: student_cdf
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.
-
Input:
student_cdf(5,2)
Input:
- Input:
student_cdf(5,-2,2)
Output:
The inverse distribution function for Student’s distribution: student_icdf
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.
Input:
student_icdf(5,0.95)
Output:
The upper tail cumulative function for Student’s distribution: UTPT
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.
Input:
UTPT(5,2)
Output: