Question

For questions 1,3 and 4, you must name the event, state what distribution you are using and write out the form of the equation to calculate the probability (you can use Sigma notation if you wish). However, you can calculate your answers using MATLAB. A wave energy converter (WEC) is set up to power a lighthouse. It is known that 420 waves can be expected to pass the WEC every hour. Furthermore the average height of the wave is known to be 3m with a standard deviation of 1.05m. 1) In order to generate energy, the waves must reach at least 1.5m what is the probability that a randomly observed wave can generate electricity? Assuming that the wave height is normally distributed 2) How many waves per hour are expected to generate electricity? 3) What is the probability of obtaining at least 420 waves in an hour? 4) To power the lighthouse all day, there must be at least 8 individual hours that contain 420 waves. What is the probability that the WEC generates enough power?

Answer 1

(1) Here the expected number of waves expected to pass the WEC every hour = 420

Average height of the wave = 3m

Standard deviation of the wave = 1.05 m

(1) Pr(x > 1.5 m ; 3m ; 1.05m ) = 1 - Pr(x < 1.5m ; 3m ; 1.05m)

Z = (1.5 - 3)/1.05 = -1.4286

Pr(x > 1.5 m ; 3m ; 1.05m ) = 1 - Pr(x < 1.5m ; 3m ; 1.05m) = 1 - Pr(Z < -1.4286) = 1 - 0.0766 = 0.9234

(2) Here expected number of waves per hour expected to generate electricity = 420 * 0.9234 = 387.845 or 388

(3) Here poisson parameter $\lambda$ = 420 waves per hour

Pr(X < = 420 ; 420) = 1 - Pr(X < 420 ; 410) = 1 - 0.4935 = 0.5065

(4) Here the distribution is binomial where p = 0.5065 and n = 24

Pr(X > = 8 ; 24 ; 0.5065) = 1 - BINOMCDF (x < 8 ; 24 ; 0.5065) = 1 - 0.0278 = 0.9723