HP 15C: Weibull Distribution Calculations

HP 15C:  Weibull Distribution Calculations

Introduction

The Weibull probability density distribution function is:

f(x) = (b / Θ) * (x / Θ)^(b-1) * exp(-(x / Θ)^b)

with the lower tail cumulative distribution of (-∞ to x):

Area = 1 - exp(-(x / Θ)^b)

The area function tells us what is the probability a device lasts no more than x time units.  

Area = 1 - Survival

The survival function is the probability a device lasts more than x time units.

Survival = exp(-(x / Θ)^b)

Generally, the higher Θ is, the flatter the Weibull Distribution curve.  

What follows are four calculations regarding the Weibull Distribution.  In the following programs, store the following values first prior to running the programs:

R0 = x

R1 = b

R2 = Θ

Use whatever labels you like.  

HP 15C Program:  Lower Tail Probability - Weibull Distribution

CDF = 1 - exp(-(x/Θ)^b)

Keys:

LBL B

1

RCL 0

RCL÷ 2

RCL 1

y^x

CHS

e^x

-

RTN

Key Codes:

42, 21,12

1

45, 0

45, 10, 2

45, 1

14

16

12

30

43, 32

Example:  

b = 1.96, Θ = 420

x = 300, result:  0.4038

x = 400, result:  0.5970

x = 500, result:  0.7552

HP 15C Program:  Failure Rate - Weibull Distribution

FR = b/Θ * (x/Θ)^(b-1) 

Keys:

LBL C

RCL 1

RCL÷ 2

RCL 0

RCL÷ 2

RCL 1

1

-

y^x

*

RTN

Key Codes:

42, 21, 13

45, 0

45, 10, 2

45, 0

45, 10, 2

45, 1

1

30

14

20

43, 32

Example:  

b = 1.96, Θ = 420

x = 300, result:  0.0034

x = 400, result:  0.0045

x = 500, result:  0.0055

HP 15C Program:  Mean of a Weibull Distribution

µ = (1/b)! * Θ

Keys:

LBL D

RCL 1

1/x

x!

RCL× 2

RTN

Key Codes:

42, 21, 14

45, 1

15

42, 0

45, 20, 2

43, 32

Example:  

b = 1.96, Θ = 420

Result:  373.3720

HP 15C Program:  Standard Deviation of a Weibull Distribution

σ = Θ * √((2/b)! - (1/b)!^2)

Keys:

LBL E

2

RCL 1

÷

x!

RCL 1

1/x

x!

x^2

-

√

RCL× 2

RTN

Key Codes:

42, 21, 15

2

45, 1

10

42, 0

45, 1

15

42, 0

43, 11

30

11

45, 20, 2

43, 32

Example:

b = 1.96, Θ = 420

Result:  198.2208

Sources:

HP55 Statistics Programs  Hewlett Packard Company.  Cupertino, CA.  1975

Ma, Dan.  "The Weibull distribution"  Topics in Actuarial Modeling.  September 28, 2016.   https://actuarialmodelingtopics.wordpress.com/2016/09/28/the-weibull-distribution/  Last Retrieved September 20, 2022.  

Eddie

All original content copyright, © 2011-2022.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Casio fx-CG 50: Random Walk Histogram

Casio fx-CG 50:  Random Walk Histogram

Mosquitos an Inspiration, What? 

Inspiration:  

Ellenberg, Jordan.   Shape:  The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else  Penguin Books:  New York, NY. 2021

The beginning of Chapter 4, Sir Ronald Ross proposes the problem:  In a simple case, a mosquito can only travel in one of two directions, which is northeast or southwest.   The goal was to find the average distance a mosquito is from its home after a given life span.

The program TWALK would address a similar problem.  

N = number of trials

S = number of steps the mosquito (or whatever you want to imagine) walks in a single direction

Each trial calculates the final position.  The program draws a histogram of all the trails.   

TWALK gives two choices:

(-1, 1):  Each turn, the mosquito must move forward 1 step or backward 1 step

(-1, 0, 1):  Each turn, the mosquito must either move forward 1 step, move backward 1 step, or stay still for one turn. 

Drawing an Histogram Program in Casio Programming

Syntax:

S-Grph#  DrawOn, Hist, List x, 1, [ColorLinkOff], [Color] ColorLighter

# = Graph 1, 2, 3

List x = List 1 to List 26

Color Commands:  ColorLinkOff, 

Colors: One of 8 eight colors

ColorLighter: the fill of the color is lighter than the bar

The color commands are optional.  If you are using a Casio fx-9750GIII, leave of the color commands.

Casio fx-CG50 Program TWALK

Text:

'ProgramMode:RUN

"EWS 2022-08-04"

"RANDOM WALK 2.0"

"NO. TRIALS"?->N

N->Dim List 26

"NO. STEPS"?->S

Menu "STEPS","_(-)_1,1",2,"-1,0,1",3

Lbl 2:2->F:Goto 1

Lbl 3:3->F:Goto 1

Lbl 1

For 1->I To N

F=2=>Sum (2*RanInt#(0,1,S)-1)->List 26[I]

F=3=>Sum (RanInt#((-)1,1,S))->List 26[I]

Next

"POSITIONS LIST:"Disps

List 26Disps

(-)S->H Start

1->H pitch

S-Gph1 DrawOn,Hist,List 26,1,ColorLinkOff,Blue ColorLighter 

DrawStat

On the Screen:

"EWS 2022-08-04"

"RANDOM WALK 2.0"

"NO. TRIALS"?→N

N→Dim List 26

"NO. STEPS"?→S

Menu "STEPS","-1,1",2,"-1,0,1",3

Lbl 2:2→F:Goto 1

Lbl 3:3→F:Goto 1

Lbl 1

For 1->I To N

F=2⇒Sum (2×RanInt#(0,1,S)-1)→List 26[I]

F=3⇒Sum (RanInt#(-1,1,S))→List 26[I]

Next

"POSITIONS LIST:"◢

List 26◢

-S->H Start

1->H pitch

S-Gph1 DrawOn,Hist,List 26,1,ColorLinkOff,Blue ColorLighter 

DrawStat

Examples

N = 50, S = 5

(-1, 1):  Mosquito cannot rest

(-1, 0, 1): Mosquito can rest

Wishing you an excellent day,

Eddie 

All original content copyright, © 2011-2022.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Casio fx-991EX Classwiz Tips: Normal Distribution

Casio fx-991EX Classwiz Tips:  Normal Distribution

This week I am going to show some things that can be done with the Casio fx-991EX Classwiz.  

The Normal Distribution 

The Classwiz's Distribution mode has calculations for the following probability distributions:

1.  Normal, f(x) = exp(-1/2 * ((x - μ) / σ)^2 ) / (σ * √(2 * π))

2.  Binomial

3.  Poisson

This is Mode 7.

Normal CD - Finding the Area

[ MENU ], 7: Distribution, 2: Normal CD

The CDF function calculates the area (probability) between two limits.   The lower and upper tail areas require the limits to be -∞ and +∞, respectively.  However, the Classwiz does not provide values for -∞ and +∞.  For the best estimate, I suggest using -7 and 7 are the limits.  The calculator calculates the probability from -7 to 7 to be 1.

Example:

(For all the examples, the standard values μ = 0, σ = 1)

Lower tail area to x = 3:  lower limit = -7, upper limit = 3; area:  0.9986501019

Lower limit = -2, upper limit = 2; area:  0.954499736

Upper tail from x = -1:  lower limit = 1, upper limit = 7; area:  0.1586552539

Inverse CD

[ MENU ], 7: Distribution, 3: Inverse Normal

This calculation gets the point value for a lower tail distribution (-∞ to x).

Example:

(For all the examples, the standard values μ = 0, σ = 1)

Area = 0.5; xInv:  0

Area = 0.6; xInv: 0.2533470931

Area = 0.7; xInv: 0.5244004382

Eddie

All original content copyright, © 2011-2022.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.