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 t year 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 we thus find that E (T) = β. For a new variant of the fluorescent lamps, the life expectancy β is unknown. In order to estimate β, one independently records fluorescent lamps whether they still function after one year or not. Let p = P (T> 1). c) Find an estimate and an approximate 95% confidence interval for the year when 73 of 100 fluorescent lamps were observed after one year. Show that p = e - 1 / β. Take the confidence interval for p above and find an approximate 95% 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

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