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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

RCL 0

RCL÷ 2

RCL 1

y^x

CHS

e^x

RTN

Key Codes:

42, 21,12

45, 0

45, 10, 2

45, 1

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

y^x

RTN

Key Codes:

42, 21, 13

45, 0

45, 10, 2

45, 0

45, 10, 2

45, 1

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

RCL× 2

RTN

Key Codes:

42, 21, 14

45, 1

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

RCL 1

1/x

x^2

√

RCL× 2

RTN

Key Codes:

42, 21, 15

45, 1

42, 0

45, 1

42, 0

43, 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.

Lists in Numworks - Version 19.2

Introduction

In Version 19, lists were added as an object in the Numworks calculator. We can define and name lists with any values that we want. Lists are designated by the brackets { }.

* The values can be real number, complex numbers, numbers with units, scientific constants, and combinations of those types. What is not allowed in lists are strings and matrices.

* The indexing of lists starts with 1. We can call a list's elements by the parenthesis after the list name. For example, xlist(10) recalls the 10th element of the list xlist.

* { f(k) }_k≤value generates a list of f(k) from k=1 to value, step 1. The variable can be almost any variable you want, except e and i. e is designated as the exponential constant (about 2.71828...) and i is designated as the imaginary number √-1. The limits are strictly from 1 to value.

* Once a user defined list is created, the individual values cannot be changed. Furthermore, there are no augment or delete commands. In this sense, user defined lists acts like tuples in Python.

* Lists can be recalled in the Statistics and Regression apps. The system named lists V#, X#, and Y# are updated accordingly.

* Defining a list of random values will always change the randomized values every time a user defined list is recalled.

Please note that this is for Version 19.2.

Screenshots are captured using the Numworks Emulator on July 10, 2022: www.numworks.com/emulator