The switchboard in a Minneapolis law office gets an average...

The switchboard in a Minneapolis law office gets an average of 5.5 incoming phone calls during the noon hour on Mondays. Experience shows that the existing staff can handle up to six calls in an hour. Let $X=$ the number of calls received a. Find the mean and standard deviation of $X$ b. What is the probability that the office receives at most six calls at noon on Monday? c. Find the probability that the law office receives six calls at noon. What does this mean to the law office staff who get, on average, 5.5 incoming phone calls at noon? d. What is the probability that the office receives more than eight calls at noon?

The switchboard in a Minneapolis law office gets an average of 5.5 incoming phone calls during the noon hour on Mondays. Experience shows that the existing staff can handle up to six calls in an hour. Let $X=$ the number of calls received
a. Find the mean and standard deviation of $X$
b. What is the probability that the office receives at most six calls at noon on Monday?
c. Find the probability that the law office receives six calls at noon. What does this mean to the law office staff who get, on average, 5.5 incoming phone calls at noon?
d. What is the probability that the office receives more than eight calls at noon?

Key Concepts

Poisson Distribution

The Poisson distribution is a discrete probability distribution used to model the number of events that occur in a fixed interval of time or space, given that these events happen with a known constant mean rate and independently of the time since the last event. It is particularly useful in cases where events occur rarely or randomly over continuous time or space.

Mean and Standard Deviation

For a Poisson distribution, the mean (expected value) is given by the parameter ?, which represents the average rate of occurrence. The variance of the distribution is also equal to ?, so the standard deviation is the square root of ?. These properties simplify the computation of the dispersion of the data.

Probability Mass Function (PMF)

The probability mass function for a Poisson random variable X is given by P(X = k) = (e^(-?) * ?^k) / k! for k = 0, 1, 2,... This formula provides the probability of observing exactly k events in the fixed period and is a fundamental tool in dealing with discrete event counts in Poisson processes.

Cumulative Distribution Function (CDF)

The cumulative distribution function for a Poisson variable is used to determine the probability that the variable will take a value less than or equal to a given number. It is computed by summing the probabilities given by the PMF for all k values up to the target value, which is useful for finding probabilities such as 'at most' a certain number of events.

Interpreting the Results

Interpreting probabilities derived from the Poisson distribution in context involves understanding how these probabilities reflect real-world situations of event occurrence. For example, comparing a computed probability to the capacity of a system allows for assessing whether the system is likely to be overloaded or can handle the incoming events based on the average rate.

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