## Sunday, November 13, 2022

### 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.

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