Calls to the help line of a large computer distributor...

Calls to the help line of a large computer distributor follow a Poisson distribution with a mean of 20 calls per minute. (a) What is the mean time until the one-hundredth call? (b) What is the mean time between call numbers 50 and $80 ?$ (c) What is the probability that three or more calls occur within 15 seconds?

Calls to the help line of a large computer distributor follow a Poisson distribution with a mean of 20 calls per minute.
(a) What is the mean time until the one-hundredth call?
(b) What is the mean time between call numbers 50 and $80 ?$
(c) What is the probability that three or more calls occur within 15 seconds?

Key Concepts

Poisson Process

A Poisson process is a mathematical model used to describe events that occur randomly and independently at a constant average rate over time. It is characterized by its stationary and independent increments, which imply that the number of events occurring in non-overlapping intervals are independent. This process is fundamental in many fields for modeling count-based phenomena such as arrivals, emails, or calls.

Exponential Distribution

The exponential distribution describes the time between successive events in a Poisson process. It has the key property of being memoryless, meaning that the probability of an event occurring in the future is independent of the past. This distribution is often used to model waiting times for random events occurring continuously and independently.

Gamma (Erlang) Distribution

The Gamma distribution, and its special case known as the Erlang distribution when the shape parameter is a positive integer, is used to model the waiting time until the nth event occurs in a Poisson process. It generalizes the exponential distribution, where the mean waiting time is proportional to the number of events, making it valuable for calculating expected times to reach a certain count of events.

Probability Calculation using the Poisson Distribution

The Poisson distribution provides a formula to calculate the probability of a specific number of events occurring within a fixed interval, based on the average rate of occurrence. This probability is computed by taking into account the mean number of events expected in that interval and using the Poisson probability formula, which is particularly useful when assessing the likelihood of rare events in a given timeframe.

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