Question

People arrive are arriving at a specific stop light randomly at the average rate of 4 per hour. What is the probability that more than 3 arrive between 8:00 AM and 8:30 AM on a specific day?

a).8647

b).4060

c).1429

d) .0527

Answer 1

Given that, people arrive are arriving at a specific stop light randomly at the average rate of 4 per hour. We want to find the probability that  more than 3 arrive between 8:00 AM and 8:30 AM i.e. between 30 minutes. We assume that people are arriving in poisson process with rate 4 per hour.

[ NOTE: The counting process is called a Poisson process with rates if all the following conditions hold,

1. 
2. N(t) has independent increments
3. The number of arrivals in any interval of length has distribution. ]

for this problem and interval of length     (30/60) = (1/2) Hours.

Define a random variable X is the number of people arrive in that interval, then we get  X∼Poisson(4/2) i.e. X~ Poisson(2) [ From 3rd properties of poisson process ]

PMF of X: X~ Poisson(2)

; x=0,1,2,3,......,n

Now, we want to find the probability that more than 3 people arrived in 30 minutes i.e. we want to find,

Computation of Probability:

[round to four decimal places]

Answer:- Probability of more than 3 arrive between 8:00 AM and 8:30 AM i.e. between 30 minutes is 0.1429 [round to four decimal places].

Corret Answer is " Option (c) 0.1429 ".