Exercise 1. The maximum height H of the annual flood of a river is carefully observed, because a flood greater than 6meters would be catastrophic. We model the distribution of the random variable H by a Rayleigh law whose probability density is given by X f ( x ) = - e za if x > 0 and 0 otherwise, a where a is an unknown parameter. Over an 8-year period, the following (assumed to be independent) flood heights in meters were observed: 2.1; 2.8; 1.7; 0.9; 1.8; 2.5; 2.2; 2.9. 1. Check that f defined a probability density function. 2. Give the MLE of the parameter a. 3. Give the cumulative distribution function of H. 4. Compute the probability of a disaster. 5. Give the probability to not have a flood during 100 years