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EXPONENTIAL DISTRIBUTION THE PROBABILITY DENSITY FUNCTION

Learn about the probability density function and applications of exponential distribution in analyzing random variables. Examples and calculations included.

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EXPONENTIAL DISTRIBUTION THE PROBABILITY DENSITY FUNCTION

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  1. EXPONENTIAL DISTRIBUTION THE PROBABILITY DENSITY FUNCTION • If a random variable Xis exponentially distributed with parameter  then its probability density function is given by • Mean,  = standard deviation,  = 1/ • The probability P(Xa) is obtained as follows: • If mean,  is given, find the parameter  first (see Example 2):  =1/

  2. EXPONENTIAL DISTRIBUTION Example 1: Let X be an exponential random variable with =2. Find the following:

  3. 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.

  4. 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.

  5. 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.

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

  7. NOTE • Application: • Uniform distribution • Used to generate other distributions • Normal distribution • Sum of a large number of random numbers • Normal distribution is the most widely used distribution perhaps because of this property. • If a quantity is obtained by summing up some other randomly occurring quantities, then it is very likely that the sum will be normally distributed

  8. NOTE • Exponential distribution • 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!

  9. READING AND EXERCISES • Reading: pp. 277-280 • Exercises: 7.26, 7.32

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