One of four generators supplies electricity to a remote research station in the

ID: 3182001 • Letter: O

Question

One of four generators supplies electricity to a remote research station in the Arctic. When the

first generator breaks down, the resident research team has to start another of the remaining three

generators. After two generators break down, the team has to call their homebase to place an

order for replacement generators to be sent.

There are two brands of generators at the research station. The lifetime of brand A is known to

follow an exponential distribution with mean 5 months, while the lifetime of brand B is known

to follow an exponential distribution with mean 4 months. Unfortunately, the brand labels have

been removed from the generators, so the research team only knows that they have 2 generators

of brand A and 2 generators of brand B.

The team switched on a generator at random when they arrived at the research station, and will

switch on a second generator at random when the first generator breaks down. Assume that the

lifetimes of the individual generators are independent of each other.

(a) Given that the first and the second generators are of different brands, find the mean and

the variance of the time until the research team has to call their homebase to order new

generators.

(b) Find the expected time until the research team has to call their homebase to order new

generators.

Brand A: Mean (mue)=5, m (rate) =1/mue =1/5, variance = 25, Sigma = 5

Brand B: Mean(mue) = 4, m(rate)=1/mue = 1/4, variance = 16, Sigma = 4

expected time(Brand A) = mue^(-1) + rate^(-1) = 5^(-1) + (1/5)^(-1) = 1/5 + 5 = 26/5

Similarly for brand B the expected time = 1/4 + 4 = 17/4

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