Calculators & Converters

    Weibull Distribution Probability for α = 3, k = 11 & x = 9

    Weibull Distribution Calculator

    Weibull distribution example problem workout with steps & calculation summary for shape parameter α = 3, scale parameter k = 11 & x = 9 products or services to estimate the probabilty of failure or failure rate of products or services over time, along with the estimations of mean, mode, median, sample variance.

    Calculation Summary
    Shape Parameter (α) 3
    Scale Parameter (k)11
    Random variable (x)9
    P 0.1056
    Mean µ9.8228
    Median 9.735
    Mode 9.6094
    Variance σ212.7451

    Work with Steps for α = 3, k = 11 & x = 9

    Question:
    Find the probability of failure for random variable x=9 which follows the Weibull distribution with parameters α = 3 and k = 11
    Workout :
    step 1 Address the formula input parameters & values
    shape parameter α = 3
    scale parameter k = 11
    x = 9 products or services

    step 2 Find P value using k,α & x values
    f(x) = (α/k) (x/k)(α - 1)(e(-(x/k)α))
    = (3/11) x (9/11)(3 - 1) x (e(-(9/11)3))
    = (0.2727) x (0.6695) x e(-(0.8182)3)
    = (0.2727) x (0.6695) x e-(0.5477)
    = (0.2727) x (0.6695) x (0.5783)
    PDF = 0.1056

    step 3 Find Mean using k & α values
    Mean µ = k [Γ(1 + (1/α))]
    = 11 [Γ(1 + (1/3))]
    = 11 x ( Γ(1 + 0.3333))
    = 11 x (Γ(1.3333))
    = 11 x 0.893
    mean (µ) = 9.8228

    step 4 Fine Median using k & α
    Median = k [(ln(2))1/α]
    = 11 x (0.6931)(1/3)
    = 11 x (0.6931)(0.3333)
    Median = 9.735

    step 5 Find Mode using k & α
    Mode = k(α - 1/α)(1/k)
    = 11(3 - 1/3)(1/3)
    = 11 x (2/3)(0.3333)
    = 11 x (0.8736)
    Mode = 9.6094

    step 6 Find Variance using α and k values
    Variance σ2 = k2[Γ(1 + 2/α) - [ Γ(1 + 1/α)]2 ]
    = 11²[Γ(1 + 2/3) - [ Γ(1 + 1/3)]2 ]
    = 121 x [ Γ(1 + 0.6667) - ( Γ(1 + 0.3333))² ]
    = 121 x [ Γ(1.6667) - ( Γ(1.3333))² ]
    = 121 x [(0.9028) - (0.893)² ]
    = 121 x [(0.9028) - (0.7974)]
    = 121 x 0.1053
    Variance (σ2) = 12.7451

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