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The Exponential Distribution

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. Density function. μ = mean e = 2.71828.

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The Exponential Distribution

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  1. 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.

  2. The Exponential Distribution Density function μ= mean e = 2.71828

  3. The Exponential Distribution Cumulative probabilities Where x0 is some specific value of x

  4. Example: Al’s Full-Service Pump 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.

  5. .4 .3 .2 .1 Example: Al’s Full-Service Pump Exponential Distribution f(x) P(x< 2) = 1 - 2.71828-2/3 = 1 - .5134 = .4866 x 1 2 3 4 5 6 7 8 9 10 Time Between Successive Arrivals (mins.)

  6. Using Excel to ComputeExponential Probabilities The EXPONDIST function can be used to compute exponential probabilities. The EXPONDIST function has three arguments: 1stThe value of the random variable x 2nd1/m the inverse of the mean number of occurrences in an interval 3rd“TRUE” or “FALSE” we will always enter “TRUE” because we’re seeking a cumulative probability

  7. Using Excel to ComputeExponential Probabilities • Formula Worksheet

  8. Using Excel to ComputeExponential Probabilities • Value Worksheet

  9. Relationship between the Poissonand Exponential Distributions 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

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

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

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