Likethegammadistribution,thelog-normaldistribution(Exercise2.71),theWeibulldistri- bution (Exercise2.72),andtheParetodistibution(Exercise2.73),anotherdistributionfor skewed-rightvariablesisthe Gumbeldistribution, alsocalledthe typeIextreme-valuedis- tribution. Oftenusedtomodelthedistributionofthemaximumofasetofrandomvariables (suchasthehighestlevelofariveroveraperiodofayear),ithas cdf F(y; 1, 2) = exp{exp[(y

Likethegammadistribution,thelog-normaldistribution(Exercise2.71),theWeibulldistri-

bution (Exercise2.72),andtheParetodistibution(Exercise2.73),anotherdistributionfor skewed-rightvariablesisthe Gumbeldistribution, alsocalledthe typeIextreme-valuedis-

tribution. Oftenusedtomodelthedistributionofthemaximumofasetofrandomvariables

(suchasthehighestlevelofariveroveraperiodofayear),ithas cdf F(y; θ1, θ2) = exp{−exp[−(y − θ1)~θ2]}, −∞ < y < ∞, for parameters θ2 > 0 and −∞ < θ1 < ∞. Themode = θ1, mean = θ1 + 0.577θ2, andstandard deviation = 1.283θ2. Plotthisdistribution(e.g.,usingthe dgumbel function inthe VGAM package in R) forvariousvaluesof θ1 with θ2 fixed andvariousvaluesof θ2 with θ1 fixed toillustrate that θ1 is a locationparameter and θ2 is a scaleparameter (Exercise 2.48).