Question

Suppose that cracks occur on a section of highway according to a Poisson process with
rate parameter λ = 1.2 cracks per kilometre. For a randomly selected 5km section of
the road let X be the random variable representing the number of cracks.
(i) State the distribution of X.
(ii) Find E(X) and var(X).
(iii) Find P(X = 4).
(iv) Suppose a repair crew drives from the beginning of the section. Find the proba-
bility that they encounter at least one crack in the first 3 kilometres.
(v) How far must the crew drive in order to be 99% certain of finding at least one
crack?

Answer 1

Solution

Given that = 1.2 cracks/km

X be the random variable representing the number of cracks

(a) X follows poisson distribution with mean i.e. = 1.2*5 = 6

(b) E(X) = = 6

Var (X) = = 6

(c) P(X =4) = e^-6 *(6⁴/4!)

P(X=4) = 0.1339

(d) P(X>=1) = 1 - P(X<1)

= 3*1.2 = 3.6

P(X>=1) = 1 - p(X=0)

= 1 - (e^-3.6 * 1) = 0.9727