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 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:

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

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
8-12, 8-13, 8-14, 8-22