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 that ? = 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 number of anomalies exceeds its mean value by no more than one standard deviation?
Solution : Step 1 : Here X is the number of material anomalies occurring in particular region of an aircraft gas- turbine disk. And the article propose a poisson distribution X with = 4. Based on this we have to compute different probability. Step 2 a) We have to find P(X4) and P(X<4). It is given that X~P() with parameter = 4. The probability mass function (Pmf) of poisson distribution is e x P(x) = x! , x= 0,1,2,3,... So 4 x P(X4) = e X=0 x! 4 4 x = e 4 , = 4 i=0 x! = P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4) = 0.1832+0.0733+0.1465+0.1954+0.1954 = 0.6288 We can find this probability in excel also by using the function (POISSON(X,MEAN,CUMULATIVE= FALSE),then SUM(CELL NUMBERS) ). Similarly, P(X<4) = P(X4) - p(X=4) p(X=4) = 0.1954 So P(X<4) = .6288-0.1954 = 0.4334 b) We have to compute P( 4 X 8)= P(X=4)+p(X=5)+p(X=6)+p(X=7)+p(X=8) = 0.1954+0.1563+0.1042+0.0595+0.0298 =0.5452 C) Here we have to find P(X8)= 1-p(X<8) = 1- [ P(X4)+p(x=5)+P(x=6)+P(x=7)] = 1- [.6288+.1563+.1042+0.0595] = 0.0512