4. John needs to study potholes for his civil engineering project. He estimates

Question

4. John needs to study potholes for his civil engineering project. He estimates that a particular small road has an average of 1 pothole per mile and the potholes are located according to Poisson distribution. a) What is the length of the road he has to be ready to bike to be almost sure (specifically, 90% sure) he'll find at least three potholes? b) The weather today is bad, so John has decided he won't be able to bike more than 3 miles one way. He got lucky however and found two potholes after just one mile worth of biking. Given this information what is the probability of him finding at least three potholes today.

Explanation / Answer

QUeestion here the poisson parameter is = 1 pothole per meter

so here as it is asked that there should be 90 % sure that there will be three potholes in t miles

so, expected number of potholes in t miles = t * 1 =  

Pr(x 3) = 0.90

POISSON (x 3; t) > 0.90

so here the probability of three arrivals has the distribution GAMMA (3, 1/) = GAMMA(3,1)

so now we know that

GAMMA (t ; 3; 1) = 0.90

so using gamma CDF we get using GAMMACDF function

t = 5.3223 miles

(b) Here expected number of potholes in 3 miles = 3

Here there are two potholes in initial one km so now there is two miles ride still left.

Now expected number of potholes he will find in next 2 miles = 2

so to find atleast 3 potholes today he must find one more pothole in next two km. If x is the number of potholes in next 2 km.

Pr(x 1; 2) = 1 - POISSON (x = 0 ; 2) = 1 - e-2 = 1 - 0.1353 = 0.8646

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