Question

An article suggests that a Poisson process can be used to represent the occurrence of structural loads over time. Suppose the mean time between occurrences of loads is 0.5 year. For every single part, clearly define the random variable of interest using the context of this problem. (Hint: We are given that the mean time between occurrences of loads is 0.5 year. It is not the λ value to be used in Poisson process since it is the mean time between occurrences of loads. Compute the mean number of loads in one year using that information.)

(a) How many loads can be expected to occur during a 2-year period?

(b) What is the probability that more than two loads occur during a 3.5-year period?

(c) How long must a time period be so that the probability of no loads occurring during that period is at most 0.1? (Hint: Let the time interval to be t-year and define a Poisson random variable as the number of structural loads per t years.)

Answer 1

a) number of loads expected to occur during a 2-year period =2/0.5 =4

b)number of loads expected to occur during a 3.5-year period=7

probability that more than two loads occur during a 3.5-year period =P(X>2)=1-P(X<=2)

=1-(P(X=0)+P(X=1)+P(X=2))=1-(e^-7*7⁰/0!+e^-7*7¹/1!+e^-7*7²/2!)=0.9704

c)

)number of loads expected to occur during a t year period=t/0.5 =2t

hence P(no loads occurring during that period )=e^-2t <0.1

t=-ln(0.1)/2=1.1513 Years