The article "Expectation Analysis of the Probability of Failure for Water Supply Pipes"† proposed using the Poisson distribution to model the number of failures in pipelines of various types. Suppose that for cast-iron pipe of a particular length, the expected number of failures is 1 (very close to one of the cases considered in the article). Then X, the number of failures, has a Poisson distribution with μ = 1. (Round your answers to three decimal places.) (a) Obtain P(X ≤ 4) by using the Cumulative Poisson Probabilities table in the Appendix of Tables. P(X ≤ 4) =
(b) Determine P(X = 1) from the pmf formula. P(X = 1) = Determine P(X = 1) from the Cumulative Poisson Probabilities table in the Appendix of Tables. P(X = 1) =
(c) Determine P(1 ≤ X ≤ 3). P(1 ≤ X ≤ 3) =
(d) What is the probability that X exceeds its mean value by more than one standard deviation?