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
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