An installation technician for a specialized communication...

An installation technician for a specialized communication system is dispatched to a city only when three or more orders have been placed. Suppose orders follow a Poisson distribution with a mean of 0.25 per week for a city with a population of 100,000 , and suppose your city contains a population of 800,000 . (a) What is the probability that a technician is required after a one-week period? (b) If you are the first one in the city to place an order, what is the probability that you have to wait more than two weeks from the time you place your order until a technician is dispatched?

An installation technician for a specialized communication system is dispatched to a city only when three or more orders have been placed. Suppose orders follow a Poisson distribution with a mean of 0.25 per week for a city with a population of 100,000 , and suppose your city contains a population of 800,000 .
(a) What is the probability that a technician is required after a one-week period?
(b) If you are the first one in the city to place an order, what is the probability that you have to wait more than two weeks from the time you place your order until a technician is dispatched?

Key Concepts

Poisson Distribution

The Poisson distribution is a discrete probability distribution used to model the number of events occurring in a fixed interval of time or space, assuming these events happen independently and at a constant average rate. In contexts such as counting orders or occurrences, the Poisson distribution provides a framework to predict probabilities for different counts based on the given mean rate.

Rate Scaling

Rate scaling involves adjusting the parameters of a distribution when the underlying conditions, such as population size or time interval, change. In problems where events occur at a given rate per unit time or per unit population, scaling the parameter appropriately allows accurate modeling of the process under different circumstances.

Poisson Cumulative Distribution Function

The cumulative distribution function (CDF) of a Poisson distribution is used to compute the probability that the number of events is less than or equal to a specific value. This tool is crucial when calculating probabilities for scenarios that require a minimum or maximum number of occurrences, such as triggering an action only after reaching a certain event threshold.

Exponential Distribution

The exponential distribution is the continuous counterpart to the Poisson process and is used to model the time between successive events. When events occur continuously and independently at a constant average rate, the waiting time for the next event follows an exponential distribution.

Memoryless Property

The memoryless property is a unique characteristic of the exponential distribution, stating that the probability of an event occurring in the future is independent of how much time has elapsed. This property is useful when analyzing waiting times and understanding that past waiting time does not affect the probability of future events in a Poisson process.

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