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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) = 
Γ((n+1)/2)
Γ(n/2)
nπ



1 + 
x2
n



n−1/2



 
      (5)

where recall the Gamma function (see Section 6.8.13) is defined for x>0 by

Γ(x) = 


0
ettx−1dx.

The student command finds the density function for Student’s distribution.
studentd is a synonym for student.


Example.
Input:

student(2,3)

Output:

π 
 
11
2
−1
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.


Examples.

The inverse distribution function for Student’s distribution: student_icdf

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


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.


Example.
Input:

UTPT(5,2)

Output:

0.0509697394149

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