Question

how can i do this?

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 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 year Poisson is 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

The β as 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 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 observing

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 life of all the fluorescent lamps?

Answer 1