## Sunday, September 24, 2023

### HP 15C: Error Function and Lower Tail Normal Cumulative Function

HP 15C:   Error Function and Lower Tail Normal Cumulative Function

Formulas Used

Error Function

erf(x) = 2 ÷ √π * ∫( e^(-t^2) dt, t = 0 to t = x)

CDF (Cumulative Normal Distribution Function)

CDF(x) = 1 ÷ √(2 * π) * ∫( e^(-t^2) dt, t = -∞ to t = x)

We can use the error function to calculate CDF(x):

0.5 - erf(x), for x < 0

0.5 + erf(x), for x ≥ 0

This program uses the integration function inside the program.  We will need 23 free memory registers to run the program, which will be terminated at the end of program execution.

Labels used:

B:  erf function (error function)

C:  lower tail normal CDF

4:  function used for integration

(Of course, feel free to use the labels you want, just be mindful to make the appropriate adjustments)

HP 15C Program Code:  Error Function and Lower Tail Normal Cumulative Function

Steps:  35

Bytes:  40

Step # :  Key Code :  Key

001 : 42,21,12:  LBL B

002 :  0  :   0

003 : 34 : x<>y

004 : 42,20, 4 : ∫_x_y 4

005 : 43, 32 : RTN

006 : 42,21,13 : LBL C

007 : 43,30, 2 :  TEST 2 (x<0)

008 : 43, 4, 0 : SF 0

009 : 43, 16 :  ABS

010 : 0   :   0

011 : 34 :  x<>y

012 : 2   :  2

013 : 11 :  √

014 : 10 :  ÷

015 : 42,20, 4 : ∫_y_x 4

016 : 2   :  2

017 : 10 :  ÷

018 : 48 :  .

019 : 5   :  5

020 : 34 :  x<>y

021 : 43, 6, 0 :  F?0

022 : 16  : CHS

023 : 40  :  +

024 : 43, 5, 0 : CF 0

025 : 43, 32 : RTN

026 : 42,21, 4 : LBL 4

027 : 43,11 : x^2

028 : 16 :  CHS

029 : 12 :  e^x

030 :  2  :  2

031 :  20  : ×

032 :  43,26 : π

033 :  11 :  √

034 :  10 :  ÷

035 :  43,32 : RTN

Examples

x = -0.7:  erf (N/A), CDF ≈ 0.2420

x = 0.7:  erf ≈ 0.6778, CDF ≈ 0.7580

x = 1:   erf ≈ 0.8427,  CDF ≈ 0.8413

x = 2.5  erf ≈ 0.9996,  CDF ≈ 0.9938

Eddie

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