. Let X be the number of material anomalies occurring in a particular region of an...

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” (Amer. Inst. of Aeronautics and Astronautics J., 2006: 787–793) proposes a Poisson distribution for X. Suppose μ = 4.

(a)
Compute both P(X ≤ 4) and P(X < 4).
(b)
Compute P(4 ≤ X ≤ 8).
(c)
Compute P(8 ≤ X).
(d)
What is the probability that the observed number of anomalies exceeds the expected number by no more than one standard deviation?

Solutions

Expert Solution

= 4

a) P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)

b) P(4 < X < 8) = P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8)

c) P(8 < X)

= P(X > 8)

= 1 - P(X < 8)

= 1 - (P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7))

= 1 - 0.9489

= 0.0511

d) Standard deviation = = = 2

P(X > 6) = 1 - P(X < 6)

= 1 - (P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6))

= 1 - 0.8893 = 0.1107

orchestra answered 1 year ago

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