Emergency 911 calls to a small municipality in Idaho come in...

Emergency 911 calls to a small municipality in Idaho come in at the rate of one every 2 minutes.
a. What is the expected number of 911 calls in one hour?
b. What is the probability of three 911 calls in five minutes?
c. What is the probability of no 911 calls in a five-minute period?

Key Concepts

Rate Parameter (?)

The rate parameter, often denoted as ? in the Poisson distribution, represents the average number of events per unit time or space. It is a crucial component in determining both the expected value and the probabilities associated with the occurrence of events in a specific interval. Adjusting the rate parameter for different time intervals is essential to accurately apply the Poisson model.

Expected Value in a Poisson Process

The expected value in a Poisson process is equal to the rate parameter multiplied by the length of the time interval being considered. This concept is used to predict the average number of occurrences or events over a specified period, making it a foundational element in the study and application of probability models for random events.

Time Scaling in Poisson Processes

Time scaling in Poisson processes involves adjusting the rate parameter to reflect the actual time interval of interest. Since the Poisson distribution assumes events occur at a constant mean rate, scaling the time allows one to compute probabilities over intervals that are longer or shorter than the original rate period. This is done by multiplying the base rate by the length of the new time interval.

Poisson Distribution

The Poisson distribution is a discrete probability distribution that models the number of events occurring within a fixed interval of time or space, where events occur with a known constant mean rate and independently of the time since the last event. It is commonly used to describe the probability of a given number of events happening in a fixed period of time when events occur continuously and independently.

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