The number of customers arriving per hour at a certain...

The number of customers arriving per hour at a certain automobile service facility is assumed to follow a Poisson distribution with mean $\mathrm{A}=7$. (a) Compute the probability that more than 10 customers will arrive in a 2 -hour period. (b) What is the mean number of arrivals during a 2 hour period?

The number of customers arriving per hour at a certain automobile service facility is assumed to follow a Poisson distribution with mean $\mathrm{A}=7$.
(a) Compute the probability that more than 10 customers will arrive in a 2 -hour period.
(b) What is the mean number of arrivals during a 2 hour period?

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Key Concepts

Poisson Distribution

A Poisson distribution is a discrete probability distribution that describes the probability of a given number of events occurring in a fixed interval of time or space, assuming these events occur with a constant mean rate and independently of each other. It is widely used for modeling count data where the events occur randomly and independently.

Parameter Scaling for Time Intervals

In a Poisson process, the mean number of events scales linearly with the length of the time interval under consideration. If the mean rate per unit time is known, then for any other time period, the mean is obtained by multiplying the rate by the length of that period. This property is essential for adapting the model to different time frames.

Tail Probabilities and the Cumulative Distribution Function (CDF)

To compute the probability of observing more than a certain number of events, one typically uses the cumulative distribution function (CDF) of the Poisson distribution. This involves calculating the probability that the variable is less than or equal to a threshold and then subtracting this value from one to get the probability of exceeding the threshold. This method is crucial for analyzing outcomes in the extreme tail of the distribution.

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