The exponential distribution is frequently applied to the...

The exponential distribution is frequently applied to the waiting times between successes in a Poisson process. If the number of calls received per hour by a telephone answering service is a Poisson random variable with parameter $X=6,$ we know that the time, in hours, between successive calls has an exponential distribution with parameter $\beta=1 / 6$. What is the probability of waiting more than 15 minutes between any two successive calls?

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Key Concepts

Interarrival Times

Interarrival times refer to the span of time between consecutive events in a process. In the context of a Poisson process, these waiting times are modeled by the exponential distribution, which makes it simple to calculate probabilities and expectations associated with the timing of events due to its memoryless property.

Poisson Process

A Poisson process is a stochastic process that models the occurrence of events over time or space in which the events occur independently and at a constant average rate. One important property of a Poisson process is that the number of events in any time interval follows a Poisson distribution, and the time between consecutive events (interarrival times) follows an exponential distribution.

Exponential Distribution

The exponential distribution is a continuous probability distribution that is commonly used to model the time between events in a process where events occur continuously and independently at a constant average rate. It is characterized by its rate parameter (or its reciprocal, the mean), and the probability of observing a waiting time greater than a particular value can be computed using its survival function, which is an exponential decay function.

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