(35S) Statistical Distributions Functions

[Thomas Klemm]

Here's a program for the Student's t-distribution:

Code:

T001 LBL T       ; Student's t
T002 INPUT K     ; degrees of freedom
T003 1
T004 -
T005 2
T006 /
T007 !           ; Γ((k+1)/2) = (k-1)/2 !
T008 RCL K
T009 2
T010 /
T011 1
T012 -
T013 !           ; Γ(k/2) = (k/2 - 1)!
T014 /
T015 PI
T016 RCL* K
T017 SQRT
T018 /
T019 STO G
T020 RTN

I001 LBL I       ; inverse CDF ( CDF -- X )
I002 INPUT A     ; area
I003 0.5
I004 -
I005 RCL/ G
I006 STO X       ; initial guess
I007 XEQ C003    ; calulate cdf(X)
I008 RCL- A      ; we want this difference to become 0
I009 RCL X
I010 STO T
I011 R↓
I012 XEQ D001    ; derivative at X = pdf(X)
I013 /
I014 STO- X      ; improved guess
I015 ABS
I016 RCL E       ; small like 1e-6
I017 x<y?        ; close enough?
I018 GTO I007    ; not yet
I019 RCL X
I020 RTN

C001 LBL C       ; CDF
C002 INPUT X
C003 0           ; lower limit
C004 RCL X       ; upper limit
C005 FN= D       ; the distribution
C006 ∫FN d T     ; integrate pdf
C007 0.5         ; we add 0.5 since
C008 +           ; the lower limit was 0
C009 RTN

P001 LBL P       ; PDF
P002 INPUT X
P003 STO T

D001 LBL D       ; Distribution: ( T -- PDF )
D002 RCL T
D003 x^2
D004 RCL/ K
D005 1
D006 +           ; 1 + t^2/k
D007 LASTx
D008 RCL+ K
D009 2
D010 /           ; (1 + k)/2
D011 y^x
D012 RCL G
D013 x<>y
D014 /
D015 RTN

You have to initialize the program with the degrees of freedom:

XEQ T ENTER
K?
7
R/S

To calculate the PDF use the program P:

XEQ P ENTER
X?
1.6
R/S
0.110665

PDF[StudentTDistribution[7],1.6]

To calculate the CDF use the program C:

XEQ C ENTER
X?
1.5
R/S
0.911351

CDF[StudentTDistribution[7],1.5]

For the next program set the accuracy to a reasonable value:

1E-6
STO E

To calculate the Inverse CDF use the program I:

XEQ I ENTER
X?
0.99
R/S
2.997951


HTH
Thomas