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

Question

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).

P(X 4) =

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?

P(X 4) =

P(X < 4) =

Explanation / Answer

Ans:

Given that

mean=4

P(x=k)=e-4*(4k/k!)

a)P(x<=4)=e-4*(40/0!+41/1!+42/2!+43/3!+44/4!)=0.6288

P(x<4)=P(x<=3)=e-4*(40/0!+41/1!+42/2!+43/3!)=0.4335

b)P(4<=x<=9)=e-4*(44/4!+45/5!+46/6!+47/7!+48/8!+49/9!)=0.5584

c)P(x>=9)=1-P(x<=8)=1-0.9786=0.0214

d)standard deviation=sqrt(4)=2

mean=4

P(x<=4+2)=P(x<=6)=0.8893

x p(x) 0 0.0183 1 0.0733 2 0.1465 3 0.1954 4 0.1954 5 0.1563 6 0.1042 7 0.0595 8 0.0298 9 0.0132 10 0.0053 11 0.0019 12 0.0006 13 0.0002 14 0.0001 15 0.0000

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