# Student’s distribution

### 8.4.8 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) =

Γ((n+1)/2)

Γ(n/2)

√

nπ

⎛
⎜
⎜
⎝

1 +

x2
n

⎞
⎟
⎟
⎠

−n−1/2

(5)

where recall the Gamma function (see Section 5.8.13) is defined for x>0 by

Γ(x) =

∫

∞
0

e−ttx−1dx.

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:

2

√

π

√

11
2

−1

2

√

2 π

· 11

which can be numerically approximated by:
Input:

evalf(student(2,3))

Output:

0.0274101222343

#### 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:

  0.949030260585

- Input:
  student_cdf(5,-2,2)
  Output:

  0.89806052117

#### 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:

2.01504837333

#### 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:

0.0509697394149