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
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