Question

The average number of flaws in a 75 meter of fiber optic cable is 2.1. suppose the manufacturing process is started.

a) What is the probability that we can make more than 25 meters until the first flaw is seen?

b) Suppose that the flaws are generally considered mild on their own. It's not until 10 flaws are seen that a cable is considered too flawed to be used. State the distribution that best models the amount of cable produced until 10 flaws are seen.

Answer 1

Here the expected number of flaws in 25 meters = 2.1 * 25/75 = 0.7

we have to find

Pr(first flaw will be seen in the first 25 meters) so here the distribution is exponential where the exponential parameter   = 2.1/75

so,

Here

f(x) = (2.1/75) e^-2.1x/75

F(x) = 1 - e^-2.1x/75

Pr(x < 25) =  1 - e^{-2.1 * 25/75} = 0.5034

(b) Here the distribution that best models the amount of cable produced until 10 flaws are seen is gamma distribution with parameter = 10 and = 75/2.1