HP 15C and TI-84 Plus CE: Weibull Distribution Parameter Calculation
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.
In today's blog, we are estimating the parameters b and Θ given the number of data points, N, and data points (time periods to failure) x_i. For the HP 15C program, which is modeled after the HP 55 program (see source below).
The Process
Each x_i is sorted in ascending order. Then transform the following data:
x' = ln x
α = (K - 0.3) ÷ (N + 0.4), K = 1, 2, 3, ... , N
y' = ln( ln( 1 ÷ (1 - α)))
Enter each point (x', y'), and perform a linear regression analysis.
Then:
b = slope
Θ = e^-(intercept ÷ slope)
HP 15C Program: Weibull Distribution - Parameter Determination
Line #; Key; Code
001; LBL A; 42, 21, 11
002; 1; 1
003; STO 0; 44, 0
004; CLΣ; 43, 32
005; R/S; 31
006; STO 1; 44, 1
007; LBL 9; 42, 21, 9
008; R/S; 31
009; LN; 43, 12
010; RCL 0; 45, 0
011; . ; 48
012; 3 ; 3
013; - ; 30
014; RCL 1; 45, 1
015; . ; 48
016; 4 ; 4
017; + ; 40
018; ÷ ; 10
019; 1 ; 1
020; STO+ 0; 44, 40, 0
021; x<>y ; 34
022; - ; 30
023; 1/x ; 15
024; LN ; 43, 12
025; LN ; 43, 12
026; x<>y ; 34
027; Σ+ ; 49
028; GTO 9; 22, 9
029; LBL B; 42, 21, 12
030; L.R.; 42, 49
031; x<>y ; 34
032; R/S ; 31
033; ÷ ; 10
034; CHS; 16
035; e^x; 12
036; RTN; 43, 32
1. Execute label A.
2. Enter N, the number of data points, then press the R/S key.
3. Enter each x_i in ascending order, press R/S key in between each keys.
4. Execute label B. The b parameter is displayed.
5. Press R/S. The Θ parameter is displayed.
TI-84 Plus CE Program: WBFIT
Weibull Distribution - Parameter Determination
"EWS 2022-10-09"
ClrHome
Disp "WEIBULL DIST.","FIT CALCULATION"
Input "DATA LIST: ",L1
SortA(L1)
dim(L1)→N
ln(L1)→L1
N→dim(L2)
For(K,1,N)
(K-0.3)/(N+0.4)→A
ln(ln(1/(1-A)))→L2(K)
End
LinReg(a+bx) L1,L2
b→B
e^((a/b))→θ
ClrHome
Disp "1-e^((X/B)^θ)"
Disp "B:",B,"θ:",θ
The x_i data are sorted in the WBIT program.
Examples
Example 1:
Hours to failure:
{ 11000, 11056, 11379, 11821, 11956, 12403, 12526, 13000, 13380, 13663 }
N = 10
b ≈ 14.01123
Θ ≈ 12649.59071
Example 2:
Days to failure:
{ 1760, 1799, 1882, 1931, 1996, 2004, 2150 }
N = 7
b ≈ 15.22473
Θ ≈ 1993.22461
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.
