Question

For its operation, one type of electronic device has two batteries: a main battery and a secondary battery. The device works with a single battery. When the main battery reaches the end of its useful life, the secondary battery automatically operates the device. At the end of the life of the secondary battery, the device is no longer usable. Let X and Y be the lifetimes (in years) of the main and secondary batteries respectively. The variable X is distributed according to a uniform law over the interval (0, 10). The conditional distribution of the variable Y knowing X = x is an exponential distribution of mean x.

a) What is the probability that such a device will operate for more than 11 years?

b) We observe 36 devices of the same type, independent of each other. What is, approximately, the probability that the average lifespan of the 36 devices exceeds (11 + 1) years?

Answer 1

The expected life of the primary battery is = 10/2 = 5 years.

E[X] = 5 years

var[X] = (10-0)^2/12 = 8.33 years^2

So

E[Y] = 5 years

var[Y] = 25 years^2

The expected lifetime of the device is = 10 years

The variance of the lifetime of the device is = 33.33 years^2

a) The probability that a device will operate for more than 11 years is

b) The probability that the average life span of 36 devices exceeds 12 years is

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