0.16744232
0.30232367
0.69767633
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4.162773
3.535534
0.16744232
0.30232367
0.69767633
—
—
—
4.162773
3.535534
The Weibull Distribution Calculator computes probability density, cumulative distribution, reliability, and hazard rate for the Weibull distribution. Named after Waloddi Weibull, this distribution is the workhorse of reliability engineering and survival analysis, capable of modeling increasing, decreasing, or constant failure rates through a single shape parameter.
The Weibull distribution with shape $$k > 0$$ and scale $$\lambda > 0$$ has the PDF:
$$f(x; k, \lambda) = \frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1} e^{-(x/\lambda)^k}, \quad x \geq 0$$
The CDF has a simple closed form:
$$F(x; k, \lambda) = 1 - e^{-(x/\lambda)^k}$$
The shape parameter $$k$$ determines the failure behavior. When $$k < 1$$, the hazard rate decreases over time (early failures, "infant mortality"). When $$k = 1$$, the Weibull reduces to the exponential distribution with constant hazard rate. When $$k > 1$$, the hazard rate increases (wear-out failures). This flexibility is why the Weibull is so popular in reliability analysis.
The hazard (failure) rate function is:
$$h(x) = \frac{f(x)}{1 - F(x)} = \frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}$$
This is a power law in $$x$$, which makes the Weibull distribution uniquely useful for modeling systems where the failure rate changes monotonically with age.
The mean is $$\lambda \cdot \Gamma(1 + 1/k)$$ and the median is $$\lambda (\ln 2)^{1/k}$$. The variance is $$\lambda^2 [\Gamma(1 + 2/k) - \Gamma(1 + 1/k)^2]$$. The mode is $$\lambda \left(\frac{k-1}{k}\right)^{1/k}$$ for $$k > 1$$ and 0 for $$k \leq 1$$.
The Weibull distribution is the standard model in reliability engineering for component lifetime analysis, warranty prediction, and maintenance scheduling. It is used in wind energy to model wind speed distributions, in material science for fracture strength (weakest-link theory), in hydrology for extreme rainfall modeling, and in survival analysis for patient time-to-event data.
Enter the shape parameter $$k$$, scale parameter $$\lambda$$, and a value $$x \geq 0$$. The calculator computes the PDF, CDF, reliability (survival) function, hazard rate, mean, variance, median, and mode.
The PDF is computed directly as $$(k/\lambda)(x/\lambda)^{k-1}\exp(-(x/\lambda)^k)$$. The CDF uses the exact formula $$1 - \exp(-(x/\lambda)^k)$$. The hazard rate is $$f(x)/[1-F(x)] = (k/\lambda)(x/\lambda)^{k-1}$$. The mean and variance use the gamma function approximated via Stirling's formula.
The reliability $$R(x) = 1 - F(x)$$ gives the probability of surviving beyond time $$x$$. The hazard rate at $$x$$ represents the instantaneous failure rate given survival to time $$x$$. If $$k < 1$$, the hazard decreases (good for burn-in items); if $$k > 1$$, it increases (wear-out). The scale $$\lambda$$ is approximately the 63.2nd percentile of the distribution.
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With k = 2 (increasing hazard), 30.2% of components fail before x = 3, and the hazard rate at x = 3 is 0.24. The mean lifetime is about 4.43 time units.
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With k = 1, the Weibull reduces to the exponential distribution. The hazard rate is constant at 1/λ = 0.1, and the mode is at 0 (monotonically decreasing PDF).
When $$k < 1$$, failures decrease over time (infant mortality). When $$k = 1$$, failures occur at a constant rate (random failures). When $$k > 1$$, failures increase over time (wear-out). When $$k \approx 3.6$$, the Weibull closely approximates the normal distribution.
It can model all three phases of the bathtub curve (decreasing, constant, increasing failure rates) with a single model by varying $$k$$. It has closed-form expressions for PDF, CDF, hazard, and reliability. And Weibull probability plots provide a simple graphical method for parameter estimation.
The scale parameter $$\lambda$$ is the characteristic life—the time at which 63.2% of the population has failed ($$F(\lambda) = 1 - e^{-1} \approx 0.632$$). Doubling $$\lambda$$ doubles all percentiles proportionally.
The exponential distribution is a special case of the Weibull with $$k = 1$$. The Weibull generalizes the exponential by allowing the failure rate to change over time, while the exponential assumes a constant rate.
A Weibull plot displays data on axes chosen so that Weibull-distributed data appears as a straight line. The slope gives the shape parameter $$k$$ and the intercept gives the scale $$\lambda$$. It is a standard tool for graphical estimation and goodness-of-fit assessment.
Wind speeds at a given location are often well-modeled by a Weibull distribution with $$k \approx 2$$ (known as a Rayleigh distribution). This model is used to estimate energy production, determine optimal turbine placement, and forecast power generation capacity.
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The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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