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 μ = 4. (Round your answers to three decimal places.) Compute both P(X > 4) and P(X < 4).

Answer 1

P(X > 4) = 0.37116 = 0.371 to 3 d.p

P(X < 4) = 0.43347 = 0.433 to 3 d.p

Explanation:

Poisson distribution formula is given as

P(X = x) = (e^-μ)(μˣ)/x!

P(X < x) = Σ (e^-μ)(μˣ)/x! (Summation From 0 to (x-1))

P(X > x) = Σ (e^-μ)(μˣ)/x! (Summation From (x+1) to the end of the distribution)

where μ = mean = 4

x = variable whose probability is required = 4

P(X > 4) represents a fraction of all possible outcomes more than the mean

P(X > 4) = P(X=5) + P(X=6) + P(X=7) + P(X=8) + ........ P(X=N)

P(X > 4) = 1 - P(X ≤ 4) = 1 - [P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4)]

Recall,

P(X=x) = (e^-μ)(μˣ)/x!

Computing this for each of them

P(X > 4) = 0.37116

P(X < 4) represents a fraction of all possible outcomes less than the mean

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

Recall, P(X=x) = (e^-μ)(μˣ)/x!

Computing this for each of them,

P(X < 4) = 0.43347