Let X be the number of material anomalies occurring in a par

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 9).
(c) Compute P(9 X).
(d) What is the probability that the number of anomalies does not exceed the mean value by more than one standard deviation?

Solution

a)

Using a table/technology, as the mean u = 4,

P(x<=4) = 0.628836935 = 0.629 [answer]

P(x < 4) = P(x<= 3) = 0.43347012 = 0.433 [answer]

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b)

P(4<=x<=9) = P(x<=9) - P(x<=4-1)

or

P(4<=x<=9) = P(x<=9) - P(x<=3)

Using table/technology,

P(x<=9) = 0.991867757
P(x<=3) = 0.43347012

Thus,

P(4<=x<=9) = 0.558397637 [answer]

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c)

P(x>=9) = 1 - P(x<=9 - 1)

or

P(x>=9) = 1 - P(x<=8)

Uisng table/technology,
P(x<=8) = 0.978636566

Hence,

P(x>=9) = 0.021363434 or 0.021 [answer]

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d)

The standard deviation here is the square root of the mean.

Hence, it is like asking

P(x<=4+sqrt(4)) or

P(x<=6) = 0.889326022 or 0.889 [answer]