Question

The life 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 still work after one year. In one building, 72 fluorescent lamps of the type in question are installed. When a fluorescent lamp fails it replaced by a new fluorescent lamp. It can then be shown that the number of fluorescent lamps fail during t Poisson is distributed with intensity λ = 72 / 1 . 6 = 45 per year. '

b) Find the probability that at least three fluorescent lamps failed within a 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 diene β as parameter can be written: f ( t ) = (1 / β ) e - t / β , for t ≥ 0 , The connection with the wording in the book / lecture notes is that β = 1 / λ and we so that E ( T ) = β .

For a new variant of the fluorescent lamps is the life expectancy β unknown. To estimate β detects man for n independent fluorescent 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 observed 73 out of 100 fluorescent lamps worked after one year. Show that p = e - 1 / β . Assume the confidence interval for p above and find an approximate 95% con- fiduciary interval for β . What would be a disadvantage with this confidence interval for β rather than an interval based on recording the exact life of all the fluorescent lamps?

Answer 1