Question

5. The lifetime of a car battery can be modeled as a Weibull distribution with a=0.9. a) If the probability that a battery works longer than 10 years is 0.45, find the value of the parameter λ? b) What is the time to which 75% of the battery work?

Answer 1

SOLUTION:

From given data,

5. The lifetime of a car battery can be modeled as a Weibull distribution with a=0.9.

a Weibull distribution with a=0.9

a) If the probability that a battery works longer than 10 years is 0.45, find the value of the parameter λ?

Let X be the lifetime of a car battery.

Probability that a car battery works longer than 10 years is 0.45.

=> P(X > 10) = 0.45

=> exp(-(x / λ)^a) = 0.45

=> exp(-(10 / λ)^0.9) = 0.45

=> -(10 / λ)^0.9 = log(0.45)

=> -(10 / λ)^0.9 = -0.34678

=> 10 / λ = 0.34678^1/0.9

=> 10 / λ = 0.30828

=>λ = 10 / 0.30828 = 32.438

(b) What is the time to which 75% of the battery work?

75% = 75/100 = 0.75

Let t be time to which 75% of the battery work. Then,

P(X < t) = 0.75

=> P(X > t) = 1 - 0.75 = 0.25

=> exp(-(x / λ )^a) = 0.25

=> exp(-(x / 32.438)^0.9) = 0.25

=> -(x / 32.438)^0.9 = log(0.25)

=> -(x / 32.438)^0.9 = -0.602059

=> x / 32.438= 0.602059^1/0.9

=> x / 32.438= 0.56905

=> x = 32.438* 0.56905= 18.4588