Calculators & Converters

    Weibull Distribution Probability for α = 2, k = 5 & x = 11

    Weibull Distribution Calculator

    Weibull distribution example problem workout with steps & calculation summary for shape parameter α = 2, scale parameter k = 5 & x = 11 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) 2
    Scale Parameter (k)5
    Random variable (x)11
    P 0.007
    Mean µ4.4311
    Median 4.1628
    Mode 3.5355
    Variance σ25.365

    Work with Steps for α = 2, k = 5 & x = 11

    Question:
    Find the probability of 11th failure by using Weibull distribution with parameters α = 2 and k = 5
    Workout :
    step 1 Address the formula input parameters & values
    shape parameter α = 2
    scale parameter k = 5
    x = 11 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))
    = (2/5) x (11/5)(2 - 1) x (e(-(11/5)2))
    = (0.4) x (2.2) x e(-(2.2)2)
    = (0.4) x (2.2) x e-(4.84)
    = (0.4) x (2.2) x (0.0079)
    PDF = 0.007

    step 3 Find Mean using k & α values
    Mean µ = k [Γ(1 + (1/α))]
    = 5 [Γ(1 + (1/2))]
    = 5 x ( Γ(1 + 0.5))
    = 5 x (Γ(1.5))
    = 5 x 0.8862
    mean (µ) = 4.4311

    step 4 Fine Median using k & α
    Median = k [(ln(2))1/α]
    = 5 x (0.6931)(1/2)
    = 5 x (0.6931)(0.5)
    Median = 4.1628

    step 5 Find Mode using k & α
    Mode = k(α - 1/α)(1/k)
    = 5(2 - 1/2)(1/2)
    = 5 x (1/2)(0.5)
    = 5 x (0.7071)
    Mode = 3.5355

    step 6 Find Variance using α and k values
    Variance σ2 = k2[Γ(1 + 2/α) - [ Γ(1 + 1/α)]2 ]
    = 5²[Γ(1 + 2/2) - [ Γ(1 + 1/2)]2 ]
    = 25 x [ Γ(1 + 1) - ( Γ(1 + 0.5))² ]
    = 25 x [ Γ(2) - ( Γ(1.5))² ]
    = 25 x [(1) - (0.8862)² ]
    = 25 x [(1) - (0.7854)]
    = 25 x 0.2146
    Variance (σ2) = 5.365

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