Internet Traffic Data Set 27 “Internet Traffic” includes 9000 arrivals of Internet traffic at the Digital Equipment Corporation, and those 9000 arrivals occurred over a period of 19,130 thousandths of a minute. Let the random variable x represent the number of such Internet traffic arrivals in one thousandth of a minute. It appears that these Internet arrivals have a Poisson distribution. If we want to use Formula 5-9 to find the probability of exactly 2 arrivals in one thousandth of a minute, what are the values of μ, x, and e that would be used in that formula?

Verified step by step guidance

1. Step 1: Understand the Poisson distribution formula. The probability of observing exactly x events in a fixed interval is given by the formula: P(x; μ) = (e^(-μ) * μ^x) / x!, where μ is the mean number of events in the interval, x is the number of events we are interested in, and e is the mathematical constant approximately equal to 2.718.
2. Step 2: Identify the interval and calculate μ. The problem states that there are 9000 arrivals over 19,130 thousandths of a minute. To find μ (the mean number of arrivals per one thousandth of a minute), divide the total number of arrivals (9000) by the total time in thousandths of a minute (19,130). This gives μ = 9000 / 19,130.
3. Step 3: Assign the value of x. The problem asks for the probability of exactly 2 arrivals in one thousandth of a minute. Therefore, x = 2.
4. Step 4: Recall the value of e. The constant e is a mathematical constant approximately equal to 2.718. This value is used in the Poisson formula.
5. Step 5: Substitute the values into the Poisson formula. Using the formula P(x; μ) = (e^(-μ) * μ^x) / x!, substitute μ (calculated in Step 2), x = 2, and e = 2.718 into the formula. Simplify the expression to find the probability.

Key Concepts

Poisson Distribution

The Poisson distribution is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, given a known average rate of occurrence. It is characterized by its parameter λ (lambda), which represents the average number of events in the interval. This distribution is particularly useful for modeling rare events, such as internet traffic arrivals, where events occur independently.

Mean (μ)

In the context of the Poisson distribution, the mean (μ) is equal to the average rate of occurrence of the event in the specified interval. For the given problem, μ can be calculated by dividing the total number of arrivals (9000) by the total time period in the same units (19,130 thousandths of a minute). This value is crucial for determining the probability of observing a specific number of arrivals.

Exponential Constant (e)

The constant e, approximately equal to 2.71828, is the base of the natural logarithm and is used in various mathematical contexts, including the Poisson distribution formula. In the formula for calculating probabilities in a Poisson distribution, e is raised to the power of negative μ, which helps in determining the likelihood of observing a certain number of events (x) in a given interval. Understanding how e functions in this context is essential for applying the formula correctly.