2. People enter a large room at the rate of 1.5 person per minute. Arrivals are

Question

2. People enter a large room at the rate of 1.5 person per minute. Arrivals are randomly occurring in time. At time 0 the room is empty.

a) What is the best model to use to represent the number of people in the room? What is the corresponding model to use for representing the time between consecutive arrivals?

b) Determine the probability that 2 people are in the room at 2 minutes.)

c) Determine the probability that 1 person arrives in one millisecond time interval. Also determine the probability that no one arrives in one millisecond time interval. Employ the approximation for infinitesimal time interval.

Explanation / Answer

a. The best model to use to represent the number of people and the time beween consecutive arrivals is the Poisson distribution model

b. Probability that 2 persons are there in 2 min = prob of 1 arrival in one min = 1.5^1 *e^-1.5 /1! = 0.334 as per poisson prob function which says f(x) = u^x *e^-u/x! where f(x) is the prob of x ocurrences in an interval and u is the mean number of ocurrences

c. So if 1 person in one millisecond time interval , i.e 1000 people arrive in a second and 60000 people in 1 min

so the prob = 1.5^60000 * e^-1.5 / 60000! ~ 0

prob that no one arrives in one second interval = 0 arrivals in one min so prob = 1.5^0 * e^-1.5 / 0! = 0.223

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