Patients arrive at an emergency department according to a...

Patients arrive at an emergency department according to a Poisson process with a mean of 6.5 per hour.
(a) What is the mean time until the tenth arrival?
(b) What is the probability that more than 20 minutes is required for the third arrival?

Key Concepts

Gamma Distribution

The Gamma distribution represents the sum of multiple independent exponential random variables. When considering the time until the k-th event in a Poisson process, the waiting time is Gamma distributed with a shape parameter equal to k and the original rate parameter. This property is critical when calculating mean waiting times for multiple events.

Cumulative Distribution Function (CDF) Approach for Waiting Time Problems

For problems involving the probability of waiting more than a specific time for a certain number of events, one can use the cumulative distribution function of the Gamma distribution. Alternatively, by leveraging the relationship between the counting process of the Poisson model and the waiting times, probabilities can be computed in terms of the number of events occurring within a given time interval.

Poisson Process

A Poisson process is a stochastic process used to model a sequence of events occurring randomly over time. It is characterized by a constant average rate of occurrence and the property that events in non-overlapping time intervals occur independently, making it suitable for modeling incidents such as arrivals in a queue or occurrences in time.

Exponential Distribution

In the context of a Poisson process, the time between consecutive events is modeled by an exponential distribution. This distribution is defined by a rate parameter and has the unique 'memoryless' property, meaning that the probability of an event occurring in the future is independent of the elapsed time already spent waiting.

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