Show full work Suppose undergraduate students arrive to the

Show full work

Suppose undergraduate students arrive to the main entrance of a hall according to a Poisson process having a rate of 1.5 undergraduate students per minute. What is the probability that 12 students enter the hall during a 10-minute period? On average, how many students enter the hall during 1-hour period? What is the probability that is more than 45 seconds between two student arrivals to the hall? Suppose we are observing the main entrance of the hall and 60 seconds have elapsed since we started observing, and 3 students have arrived in that time period. (part d and e) What is the probability that five students will arrive during first 2 minutes that we observe the main entrance of the hall? What is the expected number of students that will arrive to the hall during the first two minutes that we observe the main entrance?

Solution

Possion Distribution
PMF of P.D is = f ( k ) = e- x / x!
Where   
= parameter of the distribution.
x = is the number of independent trials

a)
rate of 1.5 a minute, for 10 minute the rate is 10*1.5 = 15
P( X = 12 ) = e ^-15 * 15^12 / 12! = 0.0829
b)
rate of 1.5 a minute, for 60 minute the rate is 60*1.5 = 90

d)
rate of per student is in seconds 3 students entered class,
i.e for on an average for every 20 seconds 1 student entered class

in 2 minuetes the average count of student be = 120/20 = 6 students

P( X = 5 ) = e ^-6 * 6^5 / 5! = 0.1606
e)
in 2 minuetes the average count of student be = 120/20 = 6 students