Mean Waiting Time in a Poisson Process

What is the mean waiting time until the next particle is emitted?

A radioactive mass emits particles according to a Poisson process at a mean rate of 3 particles per minute. At some point, a clock is started.

Mean Waiting Time:

The mean waiting time until the next particle is emitted in a Poisson process with a mean rate of 3 particles per minute is 1/3 of a minute, or approximately 20 seconds.

A Poisson process is a mathematical model often used to describe the occurrence of events in a fixed interval of time or space, where the events happen at a constant mean rate and independently of the time since the last event. In this case, the radioactive mass emits particles at a mean rate of 3 particles per minute, following a Poisson process.

One key characteristic of a Poisson process is that the waiting time until the next event follows an exponential distribution with a rate parameter equal to the mean rate of the process. Since the mean rate is 3 particles per minute in this scenario, the mean waiting time until the next particle is emitted can be calculated as the reciprocal of the mean rate, which is 1/3 of a minute or 20 seconds.

Understanding the mean waiting time in a Poisson process is crucial for various applications in fields such as telecommunications, biology, and finance, where the timing of events is essential. By knowing the mean waiting time, one can better predict when the next event is likely to occur, leading to more efficient system designs and decision-making processes.