Question

The average number of passengers arriving at the Bus Station in Lancaster on Tuesday between 8:00 and 9:00 is 900.

1.To model this random phenomenon, which probability distribution is most suitable? Assume that passengers arrive at random at a constant rate throughout the hour.

2.Using the most suitable probability distribution, consider the following three statements:

I. There are 15 passengers arriving every minute on average

II. The standard deviation of the number of passengers arriving in an hour is 30

III. The probability of having 5 or more arrived customers in a period of 10 seconds is 0.11 (rounded to two decimals) Which of the statements I, II, and III are TRUE? explain

Answer 1

1) As the average number of passengers arriving is given for a fixed interval, poisson probability distribution will be the most suitable to model this phenomenon

2) All the statements I,II and III are TRUE

I) No. of passengers arriving every minute on average = 900/60 = 15

II) For poisson distribution, mean and variance are the same

Thus, standard deviation is equal to square root of mean

Standard deviation of the number of passengers arriving in an hour = √900 = 30

III) Average number of customers who arrive in a period of 10 seconds i.e (1/6) minutes is 15/6 =2.5

Probability of having 5 or more arrived customers = P(X >= 5)

= 1 - P(X < 5)

= 1 - {P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)}

= 1 -

= 1 - 0.8911

= 0.1089 ≈ 0.11