Phone calls arrive at the rate of 48 per hour at the...

Phone calls arrive at the rate of 48 per hour at the reservation desk for Regional Airways. a. Compute the probability of receiving three calls in a 5 -minute interval of time. b. Compute the probability of receiving exactly 10 calls in 15 minutes. c. Suppose no calls are currently on hold. If the agent takes 5 minutes to complete the current call, how many callers do you expect to be waiting by that time? What is the probability that none will be waiting? I d. $\mathrm{If}$ no calls are currently being processed, what is the probability that the agent can take 3 minutes for personal time without being interrupted by a call?

Phone calls arrive at the rate of 48 per hour at the reservation desk for Regional Airways.
a. Compute the probability of receiving three calls in a 5 -minute interval of time.
b. Compute the probability of receiving exactly 10 calls in 15 minutes.
c. Suppose no calls are currently on hold. If the agent takes 5 minutes to complete the current call, how many callers do you expect to be waiting by that time? What is the
probability that none will be waiting? I
d. $\mathrm{If}$ no calls are currently being processed, what is the probability that the agent can take 3 minutes for personal time without being interrupted by a call?

Key Concepts

Poisson Process

A Poisson process is a statistical model used to describe events that occur randomly and independently over time at a constant average rate. It is fundamental for modeling scenarios where events happen sporadically, such as arrivals of phone calls, radioactive emissions, or customer arrivals in a queue.

Rate Parameter (?) and Time Conversion

The rate parameter, denoted by ?, represents the average number of events occurring per unit time in a Poisson process. When dealing with time intervals that differ from the unit time, ? must be scaled appropriately by multiplying by the length of the interval, ensuring that the expected number of events is accurately reflected for that period.

Poisson Probability Mass Function

The Poisson probability mass function (PMF) is used to calculate the probability of observing a specific number of events in a fixed time interval. The PMF is given by P(X=k)= (e^(-?t) (?t)^k) / k!, where k is the number of events and ?t is the expected number of events in the interval. This function is central to determining event probabilities in applications of the Poisson process.

Exponential Distribution and Waiting Time

The exponential distribution describes the time between consecutive events in a Poisson process. It is closely related to the Poisson distribution and is especially useful for determining the probability that no events occur within a given period, which is essential for calculating waiting times or the likelihood of uninterrupted intervals in various applications.

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