Surge protection of a piece of sensitive medical equipment is analyzed. Surge in

Question

Surge protection of a piece of sensitive medical equipment is analyzed. Surge intensity is classified as low. (voltage increases by less than 5% of the nominal value. .medium. (5%- 15%), "high' (15%-40%), and detrimental, (voltage increase exceeds 40% of the nominal value). For a particular site, the probability that no surge occurs during a two-hour operation is estimated to be 0.75. It is also known that, on average, high-intensity surges occur 5 times more often than detrimental ones, medium-intensity surges occur three times more often than high-intensity ones, and low-intensity surges happen twice as often as medium-intensity ones Assume that no more than one surge will occur during the operation a) what is the probability mass function of a random variable X, where X = 0 means that no surge has occurred, X = 1 means that a low-intensity surge has occurred, X = 2 that a medium-intensity one has occurred, and X = 3 means that a high-intensity surge was detected, and finally X -4 stands for a detrimental surge? b) What is the probability that neither high-intensity nor detrimental surge occur during the operation?

Explanation / Answer

a) here probability of no surge =P(X=0)=0.75

hence P(X=1)+P(X=2)+P(X=3)+P(X=4)=1-P(X=0)=1-0.75=0.25

P(X=3)=5P(X=4) ; P(X=2)=3*P(X=3) ; P(X=1)=2*P(X=2)

hence 30*P(X=4)+15*P(X=4)+5*P(X=4)+P(X=4) =0.25

P(X=4)=0.25/51 =1/204

P(X=3) =5/204

P(X=2) =15/204

P(X=1)=30/204

hence pmf of X:

b)probability that niether high density nor detrimental surge occur =1-P(X=3)-P(X=4) =1-5/204-1/204 =198/204=0.9706

x P(x) 0 0.75 1 30/204 2 15/204 3 5/204 4 1/204

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