The annual maximum $24-$hr rainfall depth recorded by a rain gauge in St Louis, MO from$1940$~$2022$ is extracted for frequency analysis. The mean of the $24-$hr maximum precipitation is

$=71\mathrm{.}2$mm 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$)$