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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