Library / Probability distributions / PRD-11
Probability distributions
Weibull life: probability of failure by a deadline
Probability that a Weibull-distributed component has failed by time x.
- Ref
- PRD-11
- Rev
- v1
- Field
- Probability distributions
- Result
- F
F is the probability that a unit has failed by age x, and 1 − F is its reliability. At x = s the value is 1 − 1/e = 63.2 % whatever the shape k, which is why s is called the characteristic life. Use it for the fraction failed by the end of a warranty or test.
Probability that a Weibull-distributed component has failed by time x. At x = s the value is 1 − 1/e, about 63.2 %, whatever the shape.
Calculator
- Reliability, 1 − F0.5511
- Failures per 1000 units448.9
- Life by which 10 % have failed28.64 h
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- By 75 h about 44.9 % of units will have failed, or 449 in every 1000.
- At the scale life of 100 h the failed fraction would be exactly 63.2 %, whatever the shape.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| x Elapsed life x | 1 to 300 h | 2.512e-4 | 0.9993 | 0.999 | +10% | +13 % |
| s Scale life s | 10 to 300 h | 0.07916 | 1 | 0.9208 | +10% | −12.1 % |
| k Shape k | 0.5 to 5 | 0.2112 | 0.5794 | 0.3681 | +10% | −3.75 % |
Sweep of Elapsed life x
| Elapsed life x (h) | Probability of failure by x |
|---|---|
| 1 | 2.512e-4 |
| 38.38 | 0.1634 |
| 75.75 | 0.4548 |
| 113.1 | 0.7131 |
| 150.5 | 0.876 |
| 187.9 | 0.9555 |
| 225.3 | 0.9866 |
| 262.6 | 0.9966 |
| 300 | 0.9993 |
Sweep one variable
Variables and how each one contributes
Probability of failure by x
What the equation gives you. Everything else feeds it.
Shape k
The shape parameter (a plain number). Below 1 the failure rate falls with age (early failures), at 1 it is constant, and above 1 it rises (wear-out). Typical bearings have k of about 1.5 to 3. Controls how quickly the failure rate changes with age. Raising it by 10 % (to 1.98) moves the result by −3.75 %, from 0.4489 to 0.4321.
Scale life s
The scale life, in hours: the age by which 63.2 % of units have failed, whatever the shape. Stretches every time in the model in proportion. Raising it by 10 % (to 110) moves the result by −12.1 %, from 0.4489 to 0.3946.
Elapsed life x
The elapsed life, in hours, at which the result is read. Moves the age at which the density, probability or rate is evaluated. Raising it by 10 % (to 82.5) moves the result by +13 %, from 0.4489 to 0.507.
Worked example with the values above
- Scaled age: x/s = 75 / 100 = 0.75.
- Raise it to the shape: (x/s)^k = 0.59581.
- F = 1 − e^(−0.59581) = 0.44889.
Assumptions and limits
- A two-parameter Weibull with zero location: nothing fails before it is put into service.
- Failure times are independent and the units are non-repairable.
- The shape and scale come from a fit to data for the same failure mode and operating conditions.
Common mistakes
- Mixing several failure modes in one Weibull fit, which flattens the shape and hides wear-out.
- Entering the scale in a different time unit from x, such as years against hours.
- Reading the density at one age as the probability of failing at that age: it is a rate per hour.
- Treating the mean time to failure as a guaranteed life: when k > 1 many units fail well before it.
Statistics and uncertainty
The inputs are parameters, usually fitted to data, and their estimation error carries into the result. The probability or density is a smooth function of the parameters, so the sweep chart shows how sensitive it is: where the curve is steep, a small error in a parameter makes a large difference. The formula treats the parameters as exact, so it gives no confidence interval on its own.
Where it is used
- Reliability analysis of bearings, motors, capacitors, fasteners and other components.
- Warranty planning: how many units fail by the end of the warranty period.
- Choosing a replacement age from the shape of the failure rate.
Maths you need
- Exponential function: e raised to a power
- Gamma function: the generalised factorial
- Reliability: chance of surviving to a given time