Question

An offshore structure with a design life of 20 years is planned for a site where extreme wave events may occur with a return period of 100 years (i.e. the 100-year wave). The structure is designed to have a 0.99 probability of not suffering damage within its design life. Damage effects between wave events are statistically independent

(a) You found in HW 4 that the yearly probability of exceedance of the design wave height is p = 1/100 = 0.01. Here, using the information provided above and a r.v. for damage that follows a binomial distribution, find the probability of damage to the structure under a single extreme wave event (damage/exceedance).

(b) Using the damage probability (damage/year) that may be determined from your results of part (a), what is the probability of damage to the structure in the next 10 years assuming the occurrences of extreme wave events follows a Poisson process?

(c) How would your answer to part (b) differ had you assumed that extreme wave events follow the binomial distribution?

Answer 1

(a) here we use binomial distribution with parameter n=20 and p=0.01

for Binomial distribution ,P(X=r)=ⁿC_rp^r(1-p)^n-r  

probability of damage to the structure under a single extreme wave event =P(X=1)=0.1652

( using ms-excel =BINOMDIST(1,20,0.01,0)

(b) Expected number=np=10*0.01=0.1, now we use poisson distribution with paramete $\lambda$=0.1

for poisson distribution P(X=x)=exp(-$\lambda$)_$\lambda$^{^x}/x!

probability of damage to the structure under a single extreme wave event =P(X=1)=0.0905 ( using ms-excel=POISSON(1,0.1,0))

(c) for extreme events ( when probability of events is very low) we should use poisson distribution...