$3$ Simulating Random Variables $(15$ points$)$

Mannings' equation is commonly used to measure the flow in an open$-$channel:

$Q=\frac{1}{n}A(\frac{A}{P}{)}^{\frac{2}{3}}\sqrt[\phantom{2}]{S}$

Where

$Q=$ Flow Rate, $(f\frac{t^3}{s})$

$A=$ Flow Area, $(f{t}^2)$

$n=$ Manning's Roughness Coefficient

$P=$ Wetted Perimeter, $($ft$)$

$S=$ Channel Slope, $(f\frac{t}{f}t)$

It is given that an open$-$channel's flow area, channel slope, and wetted perimeter are independent random variables with following properties:

$A$ is a normal random variable with mean $30sqft$ and standard deviation $1sqft.$

$P$ is a log$-$normal random variable with mean $20ft$ with standard deviation $1ft.$

$,S$ is a uniform random variable between $0\mathrm{.}025$ and $0\mathrm{.}03.$

Additionally, for the given soil surface, the value of Mannings' Roughness Coefficient is $0\mathrm{.}025.$

Conduct $1000$ Monte$-$Carlo simulations to compute the $($a$)$ average flow rate, and $($b$)$ the median flow rate through the open channel.