Suppose that the log-ons to a computer network follow a...

Suppose that the log-ons to a computer network follow a Poisson process with an average of three counts per minute. (a) What is the mean time between counts? (b) What is the standard deviation of the time between counts? (c) Determine $x$ such that the probability that at least one count occurs before time $x$ minutes is $0.95 .$

Suppose that the log-ons to a computer network follow a Poisson process with an average of three counts per minute.
(a) What is the mean time between counts?
(b) What is the standard deviation of the time between counts?
(c) Determine $x$ such that the probability that at least one count occurs before time $x$ minutes is $0.95 .$

Key Concepts

Poisson Process

A Poisson process is a stochastic process that models the occurrence of events happening randomly over time, where the number of events in non-overlapping intervals are independent and the probability of a single event occurring in a very short interval is proportional to the length of that interval. This process is characterized by a constant average rate, making it ideal for representing random events in time such as incoming calls or arrivals.

Exponential Distribution

The exponential distribution is a continuous probability distribution that describes the time between events in a Poisson process. It is defined by a rate parameter and is memoryless, meaning that the probability of an event occurring in the future is independent of the past. This property makes it particularly suitable for modeling waiting times between random events.

Interarrival Time

Interarrival time refers to the duration between consecutive events in a Poisson process. Because these times follow an exponential distribution, both the mean and the standard deviation of the interarrival times are functions of the rate parameter. Understanding interarrival times is crucial in analyzing the temporal structure of random events.

Cumulative Distribution Function and Quantile Calculation

The cumulative distribution function (CDF) of the exponential distribution gives the probability that the waiting time until the next event is less than or equal to a specific value. This function, typically expressed as 1 - exp(-?t), is used for calculating quantiles such as the time by which a certain probability (e.g., 0.95) of an event having occurred is reached. Inverting the CDF allows for the determination of specific time thresholds corresponding to desired probabilities.

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