$1.$ a$.$ If RIq $=100$ years, what is the probability that q will be exceeded in $30$ years $($the length of a standard mortgage$)?$

b$.$ If RIq $=100($years$),$ what is the number of years t such that P$\{$Tq $<$ t$\}=0\mathrm{.}5?$

c$.$ Consider $1000$ sites on rivers around the country. Assume that the occurrence of floods at these sites is independent. What is the probability that next year there will be an annual flood at least one site that equals or exceeds the $100-$year flood discharge?Hint: The second and third questions build right off the first. That is$,$ you make use of the equation for the probability that Q$100$ is not exceeded in n years.

$2.$ Assume that annual flood discharges at a site on a river have the following cumulative distribution function:FQ$($q$)=1-$ exp$(-$Yq$),$ where q is discharge in cfs and Y is a parameter greater than zero.Note that E$[$Q$]=1/$Y where $1/$Y has units of cfs$.$Over the last $10$ years the following annual flood discharges have been measured:$15,84,175,24,92,40,59,365,126,20($ft$3/$s$)$

$($a$)$ Plot the cumulative distribution function, labeling axes. $($Note that you dont have numeric values, so you only need to plot the general shape.$)$

$($b$)$ Derive the density function for annual floods $($Hint: PDF is derivative of CDF$)$

$($c$)$ Plot the density function, labeling axes. $($Again$,$ you dont have numeric values, so you only need to plot the general shape.$)$

$($d$)$ Derive an equation giving the annual flood quantile as a function of the recurrence interval $($T$)$ and Y$.($Hint: Use equations for relationship of the CDF for the annual exceedance probability and recurrence interval of a discharge of interest$)$

$($e$)$ Using the data provided, estimate Y$.$

$($f$)$ Estimate the $100-$year annual flood discharge based on the method of moments.$($Hint: solve q using equation in $($d$))$

$($g$)$ Estimate the probability of exceeding $500$ ft$3/$s$.($Hint: use solver to find RI from $($d$)$ then determine probability$)$