Suppose that the lifetimes of the electromagnetic valve used for starting the id

Question

Suppose that the lifetimes of the electromagnetic valve used for starting the idle-up actuator of an air conditioner can be modeled by an exponential distribution with a mean of 20 million revolutions per failure. Consider a randomly selected valve.

a. Find the probability that this valve fails within the 1 million revolutions.

b. What is the expected time to failure for this valve?

c. Find the probability that this valve lasts longer than 5 milliion revolutions.

d. If two of such valves are randomly selected and tested independently, what's the probability that at least one of them lasts longer than 5 million revolutions?

Explanation / Answer

a. Using Excel function EXPONDIST, the probability that the valve fails within 1 million revolutions is as follows:

P(X<=1)=EXPONDIST(1,20,TRUE) [EXPONDIST(x,lambda, cumulative) returns the probability density function for x being the value of the function, lambda is the parameter value, for cumulative probability distribution type TRUE else type FALSE]

=0.9999

b. Expected value, E[X]=1/lambda=1/20=0.05

c. P(X>5)=1-P(X<=5)=1-POISSON(5,20,TRUE)=0.9999

d. P(Y>=1)=1-P(Y<1)=1-P(Y=0)=1-BINOMDIST(0,2,0.9999,FALSE)=0.9999

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