Question

During a typical Pennsylvania winter, I80 averages 1.6 potholes per 10miles. A certain county is responsible for repairing potholes in a 30-mile stretch of the interstate. LetXdenote the number of potholes thecounty will have to repair at the end of next winter.
(a) The distribution of the random variable X is (choose one)
(i) binomial
(ii) hypergeometric
(iii) negative binomial
(iv) Poisson.
(b) Give the expected value and variance of X.
(c)The cost of repairing a pothole is $5000. If Y denotes the county’s pothole repair expense for next winter, find the mean value and variance Y?

Answer 1

Answer:

a) (iv) Poisson.

b) E(X)=V(X)=λ=4.8

c) E(Y)=24,000

V(Y)=120,000,000

Step-by-step explanation:

We can appropiately describe this random variable with a Poisson distribution, as the probability of having a pothole can be expressed as a constant rate per mile (0.16 potholes/mile) multiplied by the stretch that correspond to the county (30 miles).

The parameter of the Poisson distribution is then:

$\lambda=0.16\cdot 30=4.8$

b) The expected value and variance of X are both equal to the parameter λ=4.8.

c) If we define Y as:

$Y=5000X$

the expected value and variance of Y are:

$E(Y)=E(5,000\cdot X)=5,000\cdot E(X)=5,000\cdot 4.8=24,000\\\\\\ V(Y)=V(5000\cdot X)=5000^2\cdot V(X)=25,000,000\cdot 4.8=120,000,000$