Vrftjkry

I would appreciate a leadverification on the following excercise.

Let $W$ denote the anual maximum wind load. The anual maximum of $W$ is distributed according to an asymptotic extreme value distribution of type I (Gumbel) with mean $mu_{W}=0.89,text{kN}/text{mm }^{2}$ and standard deviation $sigma_{W}=0.33,text{kN}/text{mm}^{2}$.

The probability density function is then given by $$fleft(wright)=frac{1}{beta}expleft[-frac{w-alpha}{beta}right]expleft[-expleft[-frac{w-alpha}{beta}right]right].$$

The mean of the Gumbel-distribution ia given by $mu_W=alpha +gamma beta$ while standard deviation is given by $sigma_W=frac{pi beta }{sqrt{6}}$. Thus, one has to calculate the location parameter $alpha$ and the scale parameter $beta$,

begin{aligned}beta=sqrt{6}frac{sigma_{W}}{pi}=0.257 & , & alpha & =mu_{W}-gammacdotbeta=0.741.end{aligned}.

However, calculating $frac{partial f}{partial w}=frac{1}{beta^{2}}expleft[frac{alpha-betaexpleft[frac{alpha-w}{beta}right]-2w}{beta}right]left(expleft[alpha/betaright]-expleft[w/betaright]right)equiv 0$, the maximum of $fleft(wright)$ is assumed at $w^ast=0.741482$, implying that $fleft(w^astright)=1.42977$ which is, of course, contrary to the definition of probabilit measures.

Now, was my approach wrong or was the parameters of excersise poorly chosen. However, I was thinking whether the appropriate distribution should be log-Gumbel as wind load cannot take values less then zero.

Thank you very much for your thoughts.

The plot of probability density function given mean $mu_W$ and standard deviation $omega_W$ using mathematica is given here.