Weibull Distribution Example

Weibull Distribution is a continuous Propability Distribution. It provides accurate failure analysis and risk predictions with extremely small samples using a simple and useful graphical plot. This worksheet help you to understand how to compute weibull distribution. The below formula used to calculate Weibull Distribution,

P(X1 < X < X₂) = e-(X₁/β)α - e-(X₂/β)α

Mean: μ = βΓ(1 + 1/α)

<>bMedian: = β(LN(2))^1/α (Natural log of 2)

mode = β(1 - 1/α)^1/α

Variance: σ² = β²[Γ(1 + 2/α) - Γ(1 + 1/α)²]
Note: Value α & β must be Positive.

Example:
Calculate the Weibull Distribution whose α & β is 2 & 5, X1 = 1, X2 = 2.
Substitute these values in the above formulas,
P(X1 < X < X2) = e-(X1/β)α - e-(X2/β)α
P(1 < X < 2) = e-(1/5)2 - e-(2/5)2
= 0.9608 - 0.8521
= 0.1087
Mean:
μ = βΓ(1 + 1/α)
= 5x Γ(1+1/2)
= 5x Γ(1.5)
= 5x 0.8864
= 4.432

Median:
= β(LN(2))^1/α
= 5x(0.6932)^(1/2)
= 5x0.8326
= 4.1629

Variance:
σ² = β²[Γ(1 + 2/α) - Γ(1 + 1/α)²]
σ² = 5²[Γ(1 + 2/2) - Γ(1 + 1/2)²]
= 25x[Γ(2)- Γ(1.5)²]
= 25x[1- 0.7857]
= 25X 0.2143
= 5.3575

Standard Deviation:
σ = √variance
= √(5.3575)
= 2.3146

When you try such calculations on your own, this weibull distribution calculator can be used to verify your results of calculations.