Question

The lifetime of a particular type of fluorescent lamp is exponentially distributed with expectation 1.6 years. Let
T be the life of a random fluorescent lamp. Assume that the lifetimes of different fluorescent lamps are independent.

a) Show that P (T> 1) = 0.535. Find P (T <1.6).
In a room, 8 fluorescent lamps of the type are installed. Find the probability that at least 6 of these fluorescent lamps will still work after one year.
In one building, 72 fluorescent lamps of the type in question are installed. When a fluorescent lamp fails, it is replaced by a new fluorescent lamp. It can then be shown that the number of fluorescent lamps that fail during tear is Poisson distributed with intensity λ = 72 / 1.6 = 45 per year.

b) Find the probability that at least three fluorescent lamps failed within one month (t = 1/12). Find the probability that at least 36 fluorescent lamps fail within one year.
The probability density of the exponential distribution formulated with the expectation value β as a parameter can be written:
f (t) = (1 / β) e − t / β, for t ≥ 0,
The relation with the wording in the book / lecture notes is that β = 1 / λ and vi
so that E (T) = β.
For a new variant of the fluorescent lamps, the life expectancy β is unknown. To estimate β registers
one uses independent lamps whether they still work after one year or not. Let p = P (T> 1).

c) Find an estimate and an approximate 95% confidence interval for p when observing that 73 out of 100 fluorescent lamps worked after one year.
Show that p = e − 1 / β.
Take the confidence interval for p above and find an approximate 95% con-
confidence interval for β.
What would be a disadvantage of this confidence interval for β rather than an interval based on recording the exact lifetimes of all the fluorescent lamps?

Answer 1