Sparagowski & Associates conducted a study of service times...

Sparagowski \& Associates conducted a study of service times at the drive-up window of fast-food restaurants. The average service time at McDonald's restaurants was $2.78 \mathrm{min}$ utes (The Cincinnati Enquirer, July 9,2000 ). Service times such as these frequently follow an exponential distribution. a. What is the probability that a customer's service time is less than 2 minutes? b. What is the probability that a customer's service time is more than 5 minutes? c. What is the probability that a customer's service time is more than 2.78 minutes?

Sparagowski \& Associates conducted a study of service times at the drive-up window of fast-food restaurants. The average service time at McDonald's restaurants was $2.78 \mathrm{min}$ utes (The Cincinnati Enquirer, July 9,2000 ). Service times such as these frequently follow an exponential distribution.
a. What is the probability that a customer's service time is less than 2 minutes?
b. What is the probability that a customer's service time is more than 5 minutes?
c. What is the probability that a customer's service time is more than 2.78 minutes?

Key Concepts

Exponential Distribution

The exponential distribution is a continuous probability distribution often used to model the time between events in a Poisson process. It is particularly useful for describing the duration of events such as service times or the time until a radioactive particle decays. A key property of the exponential distribution is that it is memoryless, meaning the probability of an event occurring in the future is independent of how much time has already elapsed.

Rate Parameter

The rate parameter, often denoted by ? (lambda), is a crucial element in defining an exponential distribution. It represents the occurrence rate of events per unit time. The rate parameter is inversely related to the mean or expected value of the distribution. In contexts where the average time between events is known, the rate parameter is computed as the reciprocal of this average time.

Cumulative Distribution Function (CDF)

The Cumulative Distribution Function (CDF) of an exponential distribution gives the probability that the event occurs before a specified time. It is defined as F(t) = 1 - e^(-?t) for t ? 0. This function is essential for finding probabilities that a random variable falls within a certain range, such as the probability of a service time being less than a particular value.

Survival Function

The survival function, also known as the tail distribution function, represents the probability that the event has not occurred by a certain time. For the exponential distribution, it is given by S(t) = e^(-?t) for t ? 0. It is used to calculate the probability that a service time exceeds a given threshold, thus complementing the cumulative distribution function.

Transcript

Please give Ace some feedback