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Weibull Distribution Calculator

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Weibull Distribution: Definition, Formula, and Applications

Weibull Distribution

Definition: The Weibull distribution is a continuous probability distribution named after Waloddi Weibull. It's particularly useful in reliability engineering and survival analysis due to its flexibility in modeling various shapes of failure rates.

Formula:The probability density function (PDF) is given by:f(x;k,λ)=kλ(xλ)k−1e−(x/λ)k,x≥0f(x;k,\lambda) = \frac{k}{\lambda}(\frac{x}{\lambda})^{k-1}e^{-(x/\lambda)^k}, \quad x \geq 0The cumulative distribution function (CDF) is:F(x;k,λ)=1−e−(x/λ)k,x≥0F(x;k,\lambda) = 1 - e^{-(x/\lambda)^k}, \quad x \geq 0

Where:

  • kk is the shape parameter (determines behavior of the failure rate)
  • λ\lambda is the scale parameter (determines the spread of the distribution)
  • xx is the random variable (must be non-negative)

Properties

  • Mean: E(X)=λΓ(1+1k)E(X) = \lambda\Gamma(1 + \frac{1}{k})
  • Variance: Var(X)=λ2[Γ(1+2k)−(Γ(1+1k))2]\text{Var}(X) = \lambda^2[\Gamma(1 + \frac{2}{k}) - (\Gamma(1 + \frac{1}{k}))^2]
  • Mode: λ(k−1k)1/k\lambda(\frac{k-1}{k})^{1/k} for k>1k > 1
  • Support: [0,∞)[0, \infty)
  • Special cases:
    • When k=1k = 1, becomes exponential distribution
    • When k=2k = 2, becomes Rayleigh distribution
    • When k=3.4k = 3.4, approximates normal distribution

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