3 A statistician investigates the number, F, of signal failures per week on a railway network. (a) The statistician assumes that signal failures occur randomly. Explain what this statement means. ] (b) State two further assumptions needed for F to be well modelled by a Poisson distribution. [2] In a random sample of 50 weeks, the statistician finds that the mean number of failures per week is 1.61, with standard deviation 1.28. (c) Explain whether this suggests that F is likely to be well modelled by a Poisson distribution. [2] Assume first that F - Po(1.61). (d) Write down an exact expression for P (F = 0). ] (e) Complete the table below to show the probabilities of different values of F, correct to three significant figures. [2] Value of F 0 >2 Probability 0.200 After further investigation, the statistician decides to use a different model for the distribution of F. In this model it is now assumed that P(F =0) is still 0.200, but that if one failure occurs, there is an increased probability that further failures occur. (f) Explain the effect of this assumption on the value of P(F = 1). [2]