4-113. Calls to a telephone system follow a Poisson...

4-113. Calls to a telephone system follow a Poisson distribution with a mean of five calls per minute. (a) What is the name applied to the distribution and parameter values of the time until the tenth call? (b) What is the mean time until the tenth call? (c) What is the mean time between the ninth and tenth calls? (d) What is the probability that exactly four calls occur within one minute? (e) If 10 separate 1 -minute intervals are chosen, what is the probability that all intervals contain more than two calls?

4-113. Calls to a telephone system follow a Poisson distribution with a mean of five calls per minute.
(a) What is the name applied to the distribution and parameter values of the time until the tenth call?
(b) What is the mean time until the tenth call?
(c) What is the mean time between the ninth and tenth calls?
(d) What is the probability that exactly four calls occur within one minute?
(e) If 10 separate 1 -minute intervals are chosen, what is the probability that all intervals contain more than two calls?

Key Concepts

Gamma (Erlang) Distribution

The gamma distribution, with the Erlang distribution as a special case when the shape parameter is an integer, models the waiting time until the k-th event in a Poisson process. It is the sum of k independent exponential random variables, providing a direct method for calculating the mean time until the tenth call, for instance.

Exponential Distribution

The exponential distribution describes the time between consecutive events in a Poisson process. It is characterized by its memoryless property, meaning that the probability of an event occurring in the future is independent of the past. This is crucial for determining the expected waiting time between individual calls.

Independence of Intervals

In a Poisson process, non-overlapping time intervals are statistically independent. This property allows the probability of events in separate time intervals to be computed individually and then combined, as in the case of calculating the probability that multiple one?minute intervals each have a certain number of calls.

Poisson Process

A Poisson process is a statistical model that describes events occurring randomly in time or space at a constant average rate, where the events occur independently of the time since the last event. This process underlies the Poisson distribution for counts and the exponential distribution for inter?arrival times, providing a framework for modeling the arrival of calls or events over time.

Poisson Distribution

The Poisson distribution is used to model the number of events occurring within a fixed interval of time when those events happen with a known constant mean rate and independently of the time since the last event. It is essential for calculating probabilities such as the chance of a given number of calls occurring in one minute.

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