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Exponential Distribution (Chapter 14). M.I.G. McEachern High School. Exponential Distribution. Exponential distributions are directly related to Poisson distributions
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Exponential Distribution(Chapter 14) M.I.G. McEachern High School
Exponential Distribution Exponential distributions are directly related to Poisson distributions • If we have a discrete (a number we can count) number of events within a specific time, then there is a continuous (a number measured) time interval between the events. • Exponential distributions calculate the probability of a certain time interval between events.
Exponential Distributions • Thus, exponential distribution problems are concerned with an event occurring or not occurring with a specific time interval. • Related to Poisson Dist., if an event does not occur, the number of events is zero, or x = 0. Thus, as the Poisson Distribution would calculate: • Moreover, if an event does not occur within the specified interval, t, then the actual interval, T, was larger. Or
Exponential Distributions • Stated again except using time intervals: The probability that an event does not occur within a specific time interval is: Where *T is the actual time (no numbers will be given) *t is the proposed specific interval divided by the time unit of lambda. * is the average rate of occurrence.
Exponential Distributions • Further, if an event does occur, this is the compliment of an event not occurring. So • Stated again except using time intervals would mean that the actual time T would be less than the specified time, t. The probability that an event occurs within a specific time interval is: Where *T is the actual time (no numbers will be given) *t is the proposed specific interval divided by the time unit of lambda. * is the average rate of occurrence.
Exponential Distributiond • Lastly, if a certain event occurs times within some interval u of time. • Then the mean or average time between events is:
Exponential Distribution Example 1: An electric motor's constant failure rate is 0.0004 failures/hr. Calculate the probability of failure for a 150 hr mission. • = 0.0004, t = 150 hr. • Further if the event is failure, and we want the probability this willhappen use: • So
Exponential Distribution Example 1: If the electric motor's constant failure rate is 0.0004 failures/hr. Calculate probability of complete success for a 150 hr mission. • = 0.0004, t = 150 hr. • Further if the event is failure, and we want the probability this does nothappen use: • So
Exponential Distribution Example 1: An electric motor's constant failure rate is 0.0004 failures/hr. What is the Mean life expectancy E(x) = ? The mean is found using Thus, 2,500 hours. So on average, we see one failure every 2500 hours.
Exponential Distribution Practice • If two customers arrive every 30 seconds on average, what is the probability of waiting less than or equal to 30 seconds for the next customer. What is the mean time between customers? • Accidents occur with a Poisson distribution at an average of 4 per week. Calculate the probability that at least two days will elapse between accidents? What is the mean time between accidents? .3189
T.O.D. A CD player has an average record of successfully operating and providing listening enjoyment for more than 5,000 hours on the average before requiring repairs. A customer is planning on buying a CD player for installation in a boat that will be taking an extended cruise that will demand 4,000 hours of play before being able to obtain repairs or routine maintenance. How reliable will the CD player be for the customer? (Assume an exponential distribution)
