Weibull Distribution Calculator
Calculate probabilities, PDF, CDF, quantiles, and statistics for Weibull distribution
Distribution Parameters
Controls the spread of the distribution (λ > 0)
Controls the shape of the distribution (k > 0)
Results
Distribution: Weibull(λ=1, k=1)
PDF: f(x) = (k/λ)(x/λ)^(k-1) × e^(-(x/λ)^k) for x ≥ 0
CDF: F(x) = 1 - e^(-(x/λ)^k) for x ≥ 0
Mean: μ = λΓ(1 + 1/k) = 0.960
Variance: σ² = λ²[Γ(1 + 2/k) - Γ²(1 + 1/k)] = 1.025
Distribution Shape
k = 1: Exponential distribution, constant hazard rate
Example Calculation
Component Lifetime Analysis
Scenario: Electronic component failure analysis
Scale Parameter (λ): 1000 hours (characteristic lifetime)
Shape Parameter (k): 2.5 (increasing failure rate)
Question: What's the probability a component lasts more than 800 hours?
Solution
P(X > 800) = 1 - F(800) = 1 - [1 - e^(-(800/1000)^2.5)]
P(X > 800) = e^(-(0.8)^2.5) = e^(-0.595) ≈ 0.552
Answer: 55.2% probability the component lasts more than 800 hours
Weibull Properties
Scale Parameter
Controls spread and characteristic value
Shape Parameter
Controls distribution shape and skewness
Versatile
Can model many distribution shapes
Common Applications
Component failure analysis, lifetime modeling
Medical survival times, duration modeling
Wind speed distributions, extreme values
Process capability, defect analysis
Fatigue analysis, stress testing
Understanding the Weibull Distribution
What is the Weibull Distribution?
The Weibull distribution is a versatile continuous probability distribution widely used in reliability engineering, survival analysis, and extreme value statistics. It was named after Swedish mathematician Waloddi Weibull who described it in 1951.
Key Parameters
- •Scale Parameter (λ): Characteristic lifetime or scale
- •Shape Parameter (k): Controls distribution shape
- •Support: x ≥ 0 (non-negative values only)
- •Flexibility: Can model various hazard rates
Shape Parameter Effects
- •k < 1: Decreasing hazard rate (infant mortality)
- •k = 1: Constant hazard rate (exponential distribution)
- •k > 1: Increasing hazard rate (wear-out failures)
- •k = 2: Rayleigh distribution
- •k ≈ 3.6: Approximately normal distribution
Reliability Tip: The Weibull distribution is particularly valuable in reliability engineering because it can model different failure patterns through its shape parameter.
Mathematical Formulas
Probability Density Function
f(x) = (k/λ)(x/λ)^(k-1) × e^(-(x/λ)^k)
Cumulative Distribution Function
F(x) = 1 - e^(-(x/λ)^k)
Mean
μ = λ × Γ(1 + 1/k)
Quantile Function
Q(p) = λ × [-ln(1-p)]^(1/k)