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LESSON 12: EXPONENTIAL DISTRIBUTION

LESSON 12: EXPONENTIAL DISTRIBUTION. Outline Finding Exponential Probabilities Expected Value, Variance and Percentiles Applications. EXPONENTIAL DISTRIBUTION THE PROBABILITY DENSITY FUNCTION.

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LESSON 12: EXPONENTIAL DISTRIBUTION

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  1. LESSON 12: EXPONENTIAL DISTRIBUTION Outline • Finding Exponential Probabilities • Expected Value, Variance and Percentiles • Applications

  2. EXPONENTIAL DISTRIBUTION THE PROBABILITY DENSITY FUNCTION • If a random variable Xis exponentially distributed with parameter  (the process rate, e.g. # per min) then its probability density function is given by • Mean,  = standard deviation,  = 1/ • The probability P(Xa) is obtained as follows:

  3. EXPONENTIAL DISTRIBUTION THE PROBABILITY DENSITY FUNCTION • If mean,  is given (e.g., length of time in min, hr etc.), find the parameter  first (see Examples 2.1, 2.2, 2.3) using the formula:  =1/

  4. EXPONENTIAL DISTRIBUTION Example 1.1: Let X be an exponential random variable with =2. Find the following:

  5. EXPONENTIAL DISTRIBUTION Example 1.2: Let X be an exponential random variable with =2. Find the following:

  6. EXPONENTIAL DISTRIBUTION Example 1.3: Let X be an exponential random variable with =2. Find the following:

  7. EXPONENTIAL DISTRIBUTION Example 1.4: Let X be an exponential random variable with =2. Find the following:

  8. EXPONENTIAL DISTRIBUTION Example 2.1: The length of life of a certain type of electronic tube is exponentially distributed with a mean life of 500 hours. Find the probability that a tube will last more than 800 hours.

  9. EXPONENTIAL DISTRIBUTION Example 2.2: The length of life of a certain type of electronic tube is exponentially distributed with a mean life of 500 hours. Find the probability that a tube will fail within the first 200 hours.

  10. EXPONENTIAL DISTRIBUTION Example 2.3: The length of life of a certain type of electronic tube is exponentially distributed with a mean life of 500 hours. Find the probability that the length of life of a tube will be between 400 and 700 hours.

  11. EXPONENTIAL DISTRIBUTIONUSING EXCEL • Excel function EXPONDIST(a,,TRUE) provides the probability P(Xa). For example, EXPONDIST(200,1/500, TRUE) = 0.3297

  12. Application • Service times, inter-arrival times, etc. are usually observed to be exponentially distributed • If the inter-arrival times are exponentially distributed, then number of arrivals follows Poisson distribution and vice versa • The exponential distribution has an interesting property called the memory less property: Assume that the inter-arrival time of taxi cabs are exponentially distributed and that the probability that a taxi cab will arrive after 1 minute is 0.8. The above probability does not change even if it is given that a person is waiting for an hour! (See problem 8-2)

  13. READING AND EXERCISES Lesson 12 Reading: Section 8-2, pp. 235-239 Exercises: 8-12, 8-13, 8-14, 8-22

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