The time to failure of fluoresent lights in a large office b

The time to failure of fluoresent lights in a large office building is eponentially distributed with failure rate of 0.03125 failures per hour.How many spare tubes must the building custodian have so that there is at least 0.95 probability of replacing all failures on a given day

Solution

Given that the time to failure of fluoresent lights in a large office building is eponentially distributed with failure rate of 0.03125 failures per hour.

Let X be an random variable the time to failuare of flurosent lights in a large office building.

X ~ Exp ( = 0.03125)

How many spare tubes must the building custodian have so that there is at least 0.95 probability of replacing all failures on a given day

That means we have given the probability and we have to find the value of x.

Symbolically we can be written as,

P(X > x) = 0.95

So here we have to find the value of x so that we have given the probability 0.95.

For that we want cumulative distribution function of Exponential distribution.

1 - P(X x) = 0.95

And P( X x ) = F(x) = 1 - e-x

1 - [ 1 - e-x ] = 0.95

1 - 1 + e-x = 0.95

e-x = 0.95

e^{(-0.03125 * x)} = 0.95

taking log on bothsides,

-0.03125 * x = log(0.95)

- 0.03125 * x = - 0.02228

x = 0.02228 / 0.03125

x = 0.71296

So P(X > 0.71296) = 0.95

Hence the answer.