Annual maximum flood data for Floyd River, Iowa is provided in the following table. Assuming the data is of log$-$normal distribution $($essentially$,$ the variable conforms to Normal distribution after a natural log transformation$),$ please calculate: the mean and standard deviation of log$-$transformed data. Based on this information, answer the following questions. $(10$ Points$)$

$($a$)$ If we are building a dam for this river with a safety design criteria of $200$ year, calculate the hydrologic risk of failure for the dam in $40$ years. $(2$ Points$)$

$($b$)$ If we can take a $10\%$ risk that the dam fails for $40$ years, what is the return year we need to design for? $($Hint: you can read off the hydrologic risk figure I pasted in my slide, or you can solve it from the hydrologic risk equation. Answer is somewhere around $400$ years$)(2$ Points$)$

$($c$)$ What are the corresponding flood magnitudes for $200$ year flood and the return year you calculate in problem $($b$)?(4$ Points$)$

$($d$)$ What is the return period of a flood event of $35,000$ cfs$?(2$ Points$)$

$\backslash$table$[[$Year$,$Discharge $($cfs$),$Year,Discharge $($cfs$)],[1935,1460,1955,2260],[1936,4050,1956,318],[1937,3570,1957,1330],[1938,2060,1958,970],[1939,1300,1959,1920],[1940,1390,1960,15,100],[1941,1720,1961,2870],[1942,6280,1962,20,600],[1943,1360,1963,3810],[1944,7440,1964,726],[1945,5320,1965,7500],[1946,1400,1966,7170],[1947,3240,1967,2000],[1948,2710,1968,829],[1949,4520,1969,17,300],[1950,4840,1970,4740],[1951,8320,1971,13,400],[1952,13,900,1972,2940],[1953,71,500,1973,5660],[1954,6250,,]]$