The Exponential Distribution The exponential distribution has a number of useful applications. For example, we can use it to describe arrivals at a car.

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Presentation transcript:

The Exponential Distribution The exponential distribution has a number of useful applications. For example, we can use it to describe arrivals at a car wash or the time it takes to load a truck.

The Exponential Distribution μ = mean e = Density function

The Exponential Distribution Cumulative probabilities Where x 0 is some specific value of x

The time between arrivals of cars at Al’s full-service gas pump follows an exponential probability distribution with a mean time between arrivals of 3 minutes. Al would like to know the probability that the time between two successive arrivals will be 2 minutes or less. Example: Al’s Full-Service Pump

Exponential Distribution x x f(x)f(x) f(x)f(x) Time Between Successive Arrivals (mins.) Example: Al’s Full-Service Pump P ( x < 2) = /3 = =.4866 P ( x < 2) = /3 = =.4866

Using Excel to Compute Exponential Probabilities The EXPONDIST function can be used to compute The EXPONDIST function can be used to compute exponential probabilities. exponential probabilities. The EXPONDIST function can be used to compute The EXPONDIST function can be used to compute exponential probabilities. exponential probabilities. The EXPONDIST function has three arguments: The EXPONDIST function has three arguments: 1 st The value of the random variable x 1 st The value of the random variable x 2 nd 1/ m 2 nd 1/ m 3 rd “TRUE” or “FALSE” 3 rd “TRUE” or “FALSE” the inverse of the mean number of occurrences in an interval in an interval we will always enter “TRUE” because we’re seeking a cumulative probability

n Formula Worksheet Using Excel to Compute Exponential Probabilities

n Value Worksheet Using Excel to Compute Exponential Probabilities

Relationship between the Poisson and Exponential Distributions The Poisson distribution provides an appropriate description of the number of occurrences per interval The Poisson distribution provides an appropriate description of the number of occurrences per interval The exponential distribution provides an appropriate description of the length of the interval between occurrences The exponential distribution provides an appropriate description of the length of the interval between occurrences

Exercise 27, p. 266 (a) (b) (c)

Exercise 27, p. 266: Continued (d) (e)