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 lambda = 4. a. What proportion of gas-turbine disks have exactly one anomaly? b. What proportion of gas-turbine disks have at least three anomalies? c. What proportion of gas-turbine disks have between one and six anomalies inclusive?

Solution

Given X~Poissonm(mean=4)

P(X=x)=(4^x)*exp(-4)/x!

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(a) P(X=1) =(4^1)*exp(-4)/1 =0.07326256

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(b) P(X>=3)=1-P(X=0)-P(X=1)-P(X=2)

=1-(4^0)*exp(-4)/1 -(4^1)*exp(-4)/1 -(4^2)*exp(-4)/2

=0.7618967

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(c) P(1<=X<=6) = P(X=1)+P(X=2)+...+P(X=6)

=(4^1)*exp(-4)/1+...+(4^6)*exp(-4)/6!

=0.8710104