Question

uppose that the longevity of a light bulb is exponential with a mean lifetime of eight years. (a) Find the probability that a light bulb lasts less than one year. (Round your answer to four decimal places.) (b) Find the probability that a light bulb lasts between six and ten years. (Round your answer to four decimal places.) (c) Seventy percent of all light bulbs last at least how long? (Round your answer to two decimal places.) yr (d) A company decides to offer a warranty to give refunds to light bulbs whose lifetime is among the lowest three percent of all bulbs. To the nearest month, what should be the cutoff lifetime for the warranty to take place? (Round your answer up to the next month.) months (e) If a light bulb has lasted seven years, what is the probability that it fails within the eighth year. (Round your answer to four decimal places.)

Answer 1

Let X be the random variable denoting the longevity of a light bulb.

Then X ~ Exp( = 1/8) { = 1/Mean}

(a)

Probability that a light bulb lasts less than one year = P(X < 1)

= 1 - exp(-1/8) (By CDF of exponential distribution, P(X < x) = 1 - exp(-x) )

= 1 - 0.8825

= 0.1175

(b)

Probability that a light bulb lasts between six and ten years = P(6 < X < 10)

= P(X > 6) - P(X > 10)

= exp(-6/8) - exp(-10/8)   (By CDF of exponential distribution, P(X > x) = exp(-x) )

= 0.4724 - 0.2865

= 0.1859

(c)

P(X .< x) = 0.70

=> 1 - exp(-x/8) = 0.70

=> exp(-x/8) = 0.30

=> (-x/8) = ln(0.30)

=> (-x/8) = -1.203973

=> x = 8 * 1.203973 = 9.63

So, Seventy percent of all light bulbs last for 9.63 years.

(d)

P(X .< x) = 0.30

=> 1 - exp(-x/8) = 0.30

=> exp(-x/8) = 0.70

=> (-x/8) = ln(0.70)

=> (-x/8) = -0.3566749

=> x = 8 * 0.3566749 = 2.8534 years

= 2.8534 * 12 35 months

The cutoff lifetime for the warranty to take place is 35 months.

(e)

Given, the light bulb has lasted seven years, the probability that it fails within the eighth year

P(X < 8 | X > 7) = P(X < 8 and X > 7) / P(X > 7) (By conditional probability definition)

= P(7 < X < 8) / P(X > 7)

= [P(X > 7) - P(X > 8)] / P(X > 7)

= [exp(-7/8) - exp(-8/8)] / exp(-7/8)

= (0.416862 - 0.3678794) / 0.416862

= 0.1175