Calculators & Converters

    Inverse Weibull distribution for α = 9, k = 6 & P = 0.75

    Weibull Distribution Calculator

    Inverse Weibull distribution example problem workout with steps & calculation summary for shape parameter α = 9, scale parameter k = 6 & P = 0.75 failure probability to estimate the lifetime of products or services over time, along with the estimations of mean, mode, median and sample variance.

    Calculation Summary
    Shape Parameter 9
    Scale Parameter6
    P 0.75
    x 6.2218
    Mean µ5.6818
    Median 5.7606
    Mode 5.922
    Variance σ20.5698

    Work with Steps for α = 9, k = 6 & P = 0.75

    Question:
    Find the inverse probability density function for Weibull distribution having the scale parameter k = 6, shape parameter α = 9 with failure probability P(x) = 0.75
    Workout :
    step 1 Address the formula input parameters & values
    shape parameter α = 9
    scale parameter k = 6
    P = 0.75

    step 2 Find x value using k = 6, α = 9 & P = 0.75
    x = k [(-ln(1 - P))(1/α)]
    = 6 x [(-ln(1 - 0.75))(1/9)]
    = 6 x [(-ln(0.25))(0.1111)]
    = 6 x [(-(-1.3863))(0.1111)]
    = 6 x (1.037)
    x = 6.2218

    step 3 Find Mean using k & α values
    Mean µ = k [Γ(1 + (1/α))]
    = 6 [Γ(1 + (1/9))]
    = 6 x ( Γ(1 + 0.1111))
    = 6 x (Γ(1.1111))
    = 6 x 0.947
    mean (µ) = 5.6818

    step 4 Fine Median using k & α
    Median = k [(ln(2))1/α]
    = 6 x (0.6931)(1/9)
    = 6 x (0.6931)(0.1111)
    Median = 5.7606

    step 5 Find Mode using k & α
    Mode = k(α - 1/α)(1/k)
    = 6(9 - 1/9)(1/9)
    = 6 x (8/9)(0.1111)
    = 6 x (0.987)
    Mode = 5.922

    step 6 Find Variance using α and k values
    Variance σ2 = k2[Γ(1 + 2/α) - [ Γ(1 + 1/α)]2 ]
    = 6²[Γ(1 + 2/9) - [ Γ(1 + 1/9)]2 ]
    = 36 x [ Γ(1 + 0.2222) - ( Γ(1 + 0.1111))² ]
    = 36 x [ Γ(1.2222) - ( Γ(1.1111))² ]
    = 36 x [(0.9126) - (0.947)² ]
    = 36 x [(0.9126) - (0.8968)]
    = 36 x 0.0158
    Variance (σ2) = 0.5698

    getcalc.com Calculators