Let X be the number of material anomalies occurring in a particular region of an aircraft gas-turbine disk. The article "Methodology for Probabilistic Life Prediction of Multiple-Anomaly Materials"� proposes a Poisson distribution for X. Suppose that ? = 4. (Round your answers to three decimal places.) (a) Compute both P(X ? 4) and P(X < 4). (b) Compute P(4 ? X ? 5). (c) Compute P(5 ? X). (d) What is the probability that the number of anomalies does not exceed the mean value by more than one standard deviation?

Answer:P(X > 4) = 0.37116 = 0.371 to 3 d.pP(X < 4) = 0.43347 = 0.433 to 3 d.pStep-by-step explanation:Poisson distribution formula is given asP(X = x) = (e^-μ)(μˣ)/x!P(X < x) = Σ (e^-μ)(μˣ)/x! (Summation From 0 to (x-1))P(X > x) = Σ (e^-μ)(μˣ)/x! (Summation From (x+1) to the end of the distribution)where μ = mean = 4x = variable whose probability is required = 4P(X > 4) represents a fraction of all possible outcomes more than the meanP(X > 4) = P(X=5) + P(X=6) + P(X=7) + P(X=8) + ........ P(X=N) P(X > 4) = 1 - P(X ≤ 4) = 1 - [P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4)]Recall,P(X=x) = (e^-μ)(μˣ)/x!Computing this for each of themP(X > 4) = 0.37116P(X < 4) represents a fraction of all possible outcomes less than the meanP(X < 4) = P(X=0) + P(X=1) + P(X=2) + P(X=3)Recall, P(X=x) = (e^-μ)(μˣ)/x!Computing this for each of them,P(X < 4) = 0.43347