Library / Probability distributions / PRD-10
Probability distributions
Weibull life: failure time density
Probability density of the time to failure of a component with a Weibull life of shape k and scale s.
- Ref
- PRD-10
- Rev
- v1
- Field
- Probability distributions
- Result
- f [1/h]
The density f gives the rate at which failures occur at age x per hour, among the original units. The area under it between two ages is the probability of failing in that interval. For k < 1 the density rises without limit as x approaches zero; for k = 1 it is the exponential density 1/s at zero; for k > 1 it starts at zero and peaks before the scale life.
Probability density of the time to failure of a component with a Weibull life of shape k and scale s. The area under the curve between two times is the probability of failing in that interval.
Calculator
- Reliability, probability of surviving to x0.5511
- Hazard rate at x0.0143 1/h
- Mean time to failure88.93 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.
- At 75 h the density is 0.007881 per hour, so about 7.88 % of the original units fail in the next 10 hours.
- The shape 1.8 is above 1, so the failure rate rises with age (wear-out).
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (1/h) | Local step | Change in result |
|---|---|---|---|---|---|---|
| k Shape k | 0.5 to 5 | 0.002428 | 0.01252 | 0.01009 | +10% | +7.64 % |
| s Scale life s | 10 to 300 h | 4.254e-17 | 0.008812 | 0.008812 | +10% | −7.47 % |
| x Elapsed life x | 1 to 300 h | 3.157e-5 | 0.008036 | 0.008005 | +10% | −3.46 % |
Sweep of Elapsed life x
| Elapsed life x (h) | Failure time density (1/h) |
|---|---|
| 1 | 4.52e-4 |
| 38.38 | 0.006999 |
| 75.75 | 0.007859 |
| 113.1 | 0.0057 |
| 150.5 | 0.003096 |
| 187.9 | 0.001328 |
| 225.3 | 4.615e-4 |
| 262.6 | 1.322e-4 |
| 300 | 3.157e-5 |
Sweep one variable
Variables and how each one contributes
Failure time density
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 +7.64 %, from 0.007881 to 0.008483 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 −7.47 %, from 0.007881 to 0.007292 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 −3.46 %, from 0.007881 to 0.007608 1/h.
Worked example with the values above
- Scaled age: x/s = 75 / 100 = 0.75.
- Power term (x/s)^(k−1) = 0.79442 and the survival factor e^(−(x/s)^k) = 0.55111.
- f = (k/s) × those = 0.0078807 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
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