Explain how to solve

Problem $1.$ Functions of Random Variables and Derived Distributions

Reference: Prof. Daniele Veneziano, MIT, Civil and Environmental Engineering $[2\mathrm{.}5$ Grade$]$

A dam is to be designed to safely retain the water in a reservoir. Let H be the maximum water level in the reservoir in a generic year, and F be the associated horizontal force acting on a $1$ meter length of the dam.

rigure $4\mathrm{.}1$

F is related to H as:

$F=c{H}^2=5H^2$

where H is in meters, and F is in kN

Suppose that H has uniform distribution between $6$ and $10$ meters, so that:

$f_H(h)=\{\begin{array}{cc}\frac{1}{4},& 6h10\\ 0,& \text{o}\text{t}\text{h}\text{e}\text{r}\text{w}\text{i}\text{s}\text{e}& \end{array}$

$($a$)$ Find the CDF of F $.$

$($b$)$ Assuming that water levels in different years are independent, plot the distribution of the maximum force in $20$ years.

$($c$)$ What practical conclusions on the design value of the horizontal force can you extract from the result in part $($b$)?$