Library / Probability distributions / PRD-12
Probability distributions
Weibull life: instantaneous failure rate
Hazard rate: the rate of failure at age x among units that are still working.
- Ref
- PRD-12
- Rev
- v1
- Field
- Probability distributions
- Result
- h [1/h]
The hazard rate h is the failure rate at age x among the units still working. It is the density divided by the reliability. For k > 1 it rises with age (wear-out), for k = 1 it is constant (random failures, the exponential model) and for k < 1 it falls (early failures). It is the quantity that tells you how risky it is to keep a unit in service.
Hazard rate: the rate of failure at age x among units that are still working. It falls with age for k < 1, is constant for k = 1 and rises for k > 1.
Calculator
- Hazard rate at the scale life s0.018 1/h
- Hazard rate at twice the age0.0249 1/h
- Ratio of hazard at 2x to hazard at x1.741
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.
- At 75 h the failure rate is 0.0143 per hour among survivors, and it keeps rising: doubling the age multiplies it by 1.74.
- A shape of 1.8 puts the unit in the wear-out region of the bathtub curve.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (1/h) | Local step | Change in result |
|---|---|---|---|---|---|---|
| s Scale life s | 10 to 300 h | 0.001979 | 0.9022 | 0.9003 | +10% | −15.8 % |
| x Elapsed life x | 1 to 300 h | 4.521e-4 | 0.04335 | 0.0429 | +10% | +7.92 % |
| k Shape k | 0.5 to 5 | 0.005774 | 0.01705 | 0.01128 | +10% | +4.45 % |
Sweep of Elapsed life x
| Elapsed life x (h) | Hazard rate (1/h) |
|---|---|
| 1 | 4.521e-4 |
| 38.38 | 0.008366 |
| 75.75 | 0.01441 |
| 113.1 | 0.01987 |
| 150.5 | 0.02496 |
| 187.9 | 0.02981 |
| 225.3 | 0.03447 |
| 262.6 | 0.03897 |
| 300 | 0.04335 |
Sweep one variable
Variables and how each one contributes
Hazard rate
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 +4.45 %, from 0.0143 to 0.01494 1/h.
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 −15.8 %, from 0.0143 to 0.01205 1/h.
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 +7.92 %, from 0.0143 to 0.01543 1/h.
Worked example with the values above
- Scaled age: x/s = 0.75.
- Raise to (k − 1): 0.75^0.8 = 0.79442.
- h = (k/s) × that = 0.0143 per hour.
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
A mean, variance or percentile is calculated from fitted parameters, and the uncertainty in those parameters passes straight through. A distribution fitted to a small sample can have parameters that are far from the truth, especially in the tails and for the variance, so quote the result together with the sample size it rests on.
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