The manager of a fast-food restaurant determines that the...

The manager of a fast-food restaurant determines that the average time that her customers wait for service is $2.5 \mathrm{min}-$ utes, (a) Find the probability that a customer has to wait more than 4 minutes. (b) Find the probability that a customer is served within the first 2 minutes. (c) The manager wants to advertise that anybody who isn't served within a certain number of minutes gets a free ham- burger. But she doesn't want to give away free hamburgers to more than 2$\%$ of her customers. What should the adver- tisement say?

The manager of a fast-food restaurant determines that
the average time that her customers wait for service is $2.5 \mathrm{min}-$
utes,
(a) Find the probability that a customer has to wait more than
4 minutes.
(b) Find the probability that a customer is served within the
first 2 minutes.
(c) The manager wants to advertise that anybody who isn't
served within a certain number of minutes gets a free ham-
burger. But she doesn't want to give away free hamburgers
to more than 2$\%$ of her customers. What should the adver-
tisement say?

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Key Concepts

Exponential Distribution

The exponential distribution is a continuous probability distribution often used to model the time between events in a Poisson process, such as customer arrivals or service times. It is characterized by a single parameter, usually the mean or rate, and is notable for its memoryless property, meaning that the likelihood of an event occurring in the future is independent of the past.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) of the exponential distribution quantifies the probability that the random variable is less than or equal to a given value. It is expressed as F(t) = 1 - e^(-?t) for t ? 0, where ? is the rate parameter. This function is fundamental in calculating the probability of events within a specified time frame.

Quantile Calculation

Quantile calculation involves finding the value below which a specified percentage of observations in a probability distribution falls. In the context of the exponential distribution, this is done by solving for the time t in the equation F(t) = p, where p is the desired cumulative probability. This method is particularly useful in applications such as setting service level thresholds or guarantees.

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