Problem $1$

The total load, $x($in tons$),$ on the roof of a building has the following probability density function:

$f_x(x)=\{\begin{array}{cc}\frac{1}{6},& 0x<mn>3</mn>\\ \frac{12}{x^3},& 3x6\\ 0& \text{o}\text{t}\text{h}\text{e}\text{r}\text{w}\text{i}\text{s}\text{e}& \end{array}$

$($a$)$ What is the expected total load? $[0\mathrm{.}5]$

$($b$)$ What is the coefficient of variation of the total load? $[0\mathrm{.}5]$

$($c$)$ What is the median total load? $[0\mathrm{.}5]$

$($d$)$ What is the probability that the total load is between $2$ and $4$ tons? $[0\mathrm{.}5]$

$($e$)$ If the roof can carry $5\mathrm{.}5$ tons before collapse, what is the probability that the roof will collapse? $[0\mathrm{.}5]$

$($f$)$ Assume that another roof with a similar design has a total load, $Y($in tons$),$ where:

$Y=2x-1$

What is the expected total load acting on the other roof? $[0\mathrm{.}25]$

$($g$)$ What is the standard deviation of the total load acting on the other roof, $Y?[0\mathrm{.}25]$

$($h$)$ The structural engineer forgot to account for one load combination. The updated probability density

function of $x$ is shown below, where $c$ is a constant. Note that the entire density function has changed

after adding the new load combination. What should the value of $c$ be$?$ Show your work. $[0\mathrm{.}5]$