1 / 23

Probability Distributions Continued

Probability Distributions Continued. Module 2d. New Topic- Failures in Time. Example problem: We have an important pump on a recycle line The packing fails on average 0.6 times/year What is the probability of the pump packing failing before 1 year?

jacie
Download Presentation

Probability Distributions Continued

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Probability Distributions Continued Module 2d

  2. New Topic- Failures in Time Example problem: • We have an important pump on a recycle line • The packing fails on average 0.6 times/year • What is the probability of the pump packing failing before 1 year? • We could also say, what is probability that the “time to failure” is less than 1 year? T.J. Harris

  3. Exponential Distribution Assume that events occur in time at an average rate  occurrences per unit time What is the probability that the first occurrence of the event happens before time “t”? Approach - • Similar to a Poisson problem but what is different? • P(event occurs before a given time) = 1 - P(event doesn’t occur during the entire time interval) T.J. Harris

  4. Exponential Distribution • Event doesn’t occur in a given time means 0 occurrences • Poisson - with occurrence rate of t in time interval t. • P(event occurs before this time) = T.J. Harris

  5. Exponential Distribution Denote continuous random variable X as the time to occurrence. Cumulative distribution function • Probability density function is Why? • Sometimes we know , the average number of failures per unit time, and sometimes we know , the mean time to failure: T.J. Harris

  6. Exponential Distribution - Notes • The time to failure is a continuous random variable • We assume that the expected failure rate is constant, and that it doesn’t increase as equipment wears out • We assume that failures are independent, and that each time increment is an independent trial • mean and variance: T.J. Harris

  7. Pump Failure Problem • If the packing fails on average 0.6 times / year, what is the chance of failure within the first year? • P(pump fails within year) • 45% chance of failure within year • We derived the Exponential distribution from the Poisson Distribution, which came from the Binomial Distribution. What troubling assumptions are we making? T.J. Harris

  8. Conditional Probability Question Given that the pump has not failed in the first 100 hours, what is the probability that the pump will fail in the next 100 hours, Given that the pump has not failed in the first 100 hours, what is the expected failure time Both questions are conditioned on specifying a survival distribution function. T.J. Harris

  9. Survival Probability: Intuitive Approach First consider the following information on a discrete random variable T.J. Harris

  10. Survival Probability: Intuitive Approach We would calculate new probabilities as Now divide numerate and denominator by n to get

  11. Survival Distribution – General Case When applied to survival data the continuous conditional distribution is known as the hazard function and is often given a special symbol The hazard function is a properly defined pdf. T.J. Harris

  12. Hazard function for exponential distribution i.e. hazard function is also exponential! T.J. Harris

  13. Hazard function for exponential distribution Apply to problem – Given that the pump has lasted for 100 hours, what is the probability that pump lasts another 100 hours before a failure T.J. Harris

  14. Exponential Distribution Memoryless property of Exponential Distribution • The probability of the component lasting for another 100 hours, given that it has functioned for 100 hours, is simply probability of it lasting 100 hours • Prior history has no influence on probability of failure when exponential distribution is used • Is this how life works? • When are we justified in using the exponential distribution and when should we avoid it? T.J. Harris

  15. How do we specify the correct form for the distribution? Suppose you have a collection of observations How do I know whether from normal, lognormal, Poisson, exponential, …? Use knowledge about physical system Could form histogram and compare to the pdf of a reference distribution Compare quantiles to reference distribution quantiles T.J. Harris

  16. T.J. Harris

  17. Core Temperature Data: Last 10K years T.J. Harris

  18. Analysis of Core Temperature Data Looks like normal distribution. Superimpose plot of pdf on this graph. Use sample mean and sample standard deviation for Superimpose the following equation on the histogram What values do we use for x? T.J. Harris

  19. Analysis of Core Temperature Data T.J. Harris

  20. Analysis of Core Temperature Data Data points lie close to red line with the exception of 1 point. Conclude that this data can be represented by normal distribution T.J. Harris

  21. Hardcover: 392 pages Publisher: Nova Science Publishers (July 30, 2008) Language: English ISBN-10: 1600217753 ISBN-13: 978-1600217753 T.J. Harris

  22. Case Study: Landfill Liner failure Problem Description Pareto Analysis Fault Analysis – Use of Conditional Probabilities Hazard Model for estimating time to failure T.J. Harris

  23. Case Study: Landfill Liner failure Hazard Failure model T.J. Harris

More Related