$(15$ points$)$ The annual maximum $24-$hr rainfall depth recorded by a rain gauge

in St Louis, MO from $19402022$ is extracted for frequency analysis. The mean of

the $24-$hr maximum precipitation is $=71\mathrm{.}2mm$ and the standard deviation is $=$

$35\mathrm{.}2(mm).$

$($a$)$ Please estimate the $100-$yr $24-$hr rainfall depth

$($b$)$ The current designed capacity for the storm drainage system in St Louis is $143$

mm for $24$ hour. What is the corresponding design return period? $($Note: assume

a Gumbel extreme value type I distribution for precipitation$)$