Question

The number of traffic accidents in a certain area follows a Poisson process with a rate of 1.5 per hour between 8:00 A.M. and 5:00 P.M. during the normal working hours in a working day. Compute the following probabilities.

1. There will be no traffic accident between 11:30 AM to 12:00 PM.
2. There will be more than 3 traffic accidents after 3:45 P.M.
3. There will be in between 15 and 18 traffic accident during the normal working hours in a working day.
4. The number of the traffic accidents during the normal working hours in a working day are within a S.D. from the mean.
5. Find the median of the number of the traffic accidents during the normal working hours in a working day.

Answer 1

1) expected number of accidents in 30 minutes between 11:30 AM to 12:00 PM =1.5*30/60=0.75

hence P( There will be no traffic accident between 11:30 AM to 12:00 PM )=P(X=0)=e^-0.75*0.75⁰/0! =0.4724

2)

expected number of accidents after 3.45 pm (in 1.25 hours)=1.5*1.25=1.875

hence There will be more than 3 traffic accidents after 3:45 P.M =P(X>3)=1-P(X<=3)=1- =0.121054

3)

number of expected accidents between 8:00 A.M. and 5:00 P.M (9 hours)=1.5*9=13.5

There will be in between 15 and 18 traffic accident = =0.285107

4)

here mean =13.5 and std deviation =√13.5 =3.674

hence 1 std deviation from mean values are =13.5-/+3.674 =9.826 to 17.174

hence P(9.826 <X<17.174)= =0.725614

5)

median of the number of the traffic accidents during the normal working hours in a working day =13