The manager of a gas station has observed that the times...

Name: Problem 80, Chapter 8: Statistics for Management and Economics
Uploaded: 2023-10-30T23:55:13-08:00
Duration: 2 min 34 s
Channel: Nicole C
Description: The manager of a gas station has observed that the times required by drivers to fill their car's tank and pay are quite variable. In fact, the times are exponentially distributed with a mean of 7.5 minutes. What is the probability that a car can complete the transaction in less than 5 minutes?

The manager of a gas station has observed that the times required by drivers to fill their car's tank and pay are quite variable. In fact, the times are exponentially distributed with a mean of 7.5 minutes. What is the probability that a car can complete the transaction in less than 5 minutes?

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Key Concepts

Exponential Distribution

The exponential distribution is a continuous probability distribution that is often used to model the time between independent events that occur at a constant average rate. This distribution is characterized by its lack of memory, meaning that the probability of an event occurring in the future is independent of how much time has already elapsed.

Mean and Rate Parameter

In the context of the exponential distribution, the mean (or expected value) is the average time between events and is inversely related to the rate parameter (lambda). Specifically, lambda is calculated as 1 divided by the mean. This relationship is crucial in converting between the average time of occurrence and the necessary parameter for the probability functions.

Cumulative Distribution Function (CDF)

The cumulative distribution function for the exponential distribution gives the probability that the random variable is less than or equal to a particular value. It is expressed as F(t) = 1 - e^(-lambda*t), where t is the time value in question and lambda is the rate parameter. This function is fundamental in solving probability problems involving waiting times or intervals between events.

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