LESSON 12 EXPONENTIAL DISTRIBUTION Outline Finding Exponential Probabilities

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

EXPONENTIAL DISTRIBUTION THE PROBABILITY DENSITY FUNCTION • If a random variable X is 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: 2

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/ 3

EXPONENTIAL DISTRIBUTION Example 1. 1: Let X be an exponential random variable with =2. Find the following: 4

EXPONENTIAL DISTRIBUTION Example 1. 2: Let X be an exponential random variable with =2. Find the following: 5

EXPONENTIAL DISTRIBUTION Example 1. 3: Let X be an exponential random variable with =2. Find the following: 6

EXPONENTIAL DISTRIBUTION Example 1. 4: Let X be an exponential random variable with =2. Find the following: 7

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. 8

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. 9

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. 10

EXPONENTIAL DISTRIBUTION USING EXCEL • Excel function EXPONDIST(a, , TRUE) provides the probability P(X a). For example, EXPONDIST(200, 1/500, TRUE) = 0. 3297 11

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 interarrival 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 82) 12

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