14   4 Petra is an engineer who works on an offshore wind farm. Petra organises the repairs of wind turbines that fail. The wind farm contract states that wind turbines must be repaired on the same day that they fail. Failures appear to happen at random, and at a fairly consistent rate over the year. In order to organise repairs, Petra uses a Poisson model with  λ = 2.8  for the number of failures per year for each wind turbine. [Data source: https://doi.org/10.1177%2F0957650915597560] Use Petra’s model to answer (a) to (f). (a) Calculate the standard deviation of the number of failures per year for a wind turbine at the wind farm. (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (b) Find the probability that a wind turbine at the wind farm fails exactly three times in a given year. (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (c) Find the probability that a wind turbine at the wind farm fails more than seven times over a three-year period. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Turn over 15   Question 4 continued (d) State fully the distribution of the time, in years, between successive failures of a wind turbine at the wind farm. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (e) Calculate the expected time, in years, between successive failures of a wind turbine at the wind farm. (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (f) Given that a particular wind turbine at the wind farm has not failed for two consecutive years, calculate the probability that it fails within the next six months. (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________