Likethegammaandlog-normaldistributions,the Weibulldistribution is positivelyskewed overthepositiverealline.Withshapeparameter k > 0 and scaleparameter > 0, its cdf is F(y;

Likethegammaandlog-normaldistributions,the Weibulldistribution is positivelyskewed overthepositiverealline.Withshapeparameter k > 0 and scaleparameter λ > 0, its cdf is F(y; λ, k) = 1 − e−(y~λ)k for y > 0.

(a) Findthe pdf.

(b) Showthatthemedian= λ[log(2)]1~k. (Themeanandstandarddeviationarealsopro-

portionalto λ. As k increases forfixed λ, themeandecreasestoward λ and themedian increases toward λ.)

(c) PlottheWeibull pdf for λ = 1 and k = 1, 2, 4 (e.g., usingthe dweibull function in R).

Describetheinfluenceof k on theshape,andexplainwhythemodeis0if k ≤ 1.

(d) TheWeibullisoftenusedinreliabilitystudiestomodelthelife-lengthofaproductbefore it fails.Overtime,thefailurerateoftheproductincreases(staysthesame,decreases)

when k > 1 (k = 1, k < 1). WhichdistributiondoestheWeibullsimplifytowhen k = 1?