1 / 24

Special Continuous Probability Distributions -Exponential Distribution -Weibull Distribution

Systems Engineering Program. Department of Engineering Management, Information and Systems. EMIS 7370/5370 STAT 5340 : PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS. Special Continuous Probability Distributions -Exponential Distribution -Weibull Distribution.

dom
Télécharger la présentation

Special Continuous Probability Distributions -Exponential Distribution -Weibull Distribution

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. Systems Engineering Program Department of Engineering Management, Information and Systems EMIS 7370/5370 STAT 5340 : PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS Special Continuous Probability Distributions -Exponential Distribution -Weibull Distribution Dr. Jerrell T. Stracener, SAE Fellow Leadership in Engineering Stracener_EMIS 7370/STAT 5340_Fall 08_09.25.08

  2. Exponential Distribution

  3. The Exponential Model - Definition • A random variable X is said to have the Exponential • Distribution with parameters , where  > 0, if the • probability density function of X is: • , for  0 • , elsewhere

  4. Properties of the Exponential Model • Probability Distribution Function • for < 0 • for  0 • *Note: the Exponential Distribution is said to be • without memory, i.e. • P(X > x1 + x2 | X > x1) = P(X > x2)

  5. Mean or Expected Value • Standard Deviation Properties of the Exponential Model

  6. Exponential Model - Example Suppose the response time X at a certain on-line computer terminal (the elapsed time between the end of a user’s inquiry and the beginning of the system’s response to that inquiry) has an exponential distribution with expected response time equal to 5 sec. (a) What is the probability that the response time is at most 10 seconds? (b) What is the probability that the response time is between 5 and 10 seconds? (c) What is the value of x for which the probability of exceeding that value is 1%?

  7. Exponential Model - Example The E(X) = 5=θ, so λ = 0.2. The probability that the response time is at most 10 sec is: or P (X>10) = 0.135 The probability that the response time is between 5 and 10 sec is:

  8. Exponential Model - Example The value of x for which the probability of exceeding x is 1%:

  9. Weibull Distribution

  10. The Weibull Probability Distribution Function • Definition - A random variable X is said to have the Weibull Probability Distribution with parameters  and , where  > 0 and  > 0, if the probability density function of is: • , for  0 • , elsewhere • Where,  is the Shape Parameter,  is the Scale • Parameter. Note: If  = 1, the Weibull reduces to • the Exponential Distribution.

  11. The Weibull Probability Distribution Function Probability Density Function f(t) 1.8 β=5.0 1.6 β=0.5 1.4 β=3.44 1.2 β=1.0 β=2.5 1.0 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 t t is in multiples of 

  12. The Weibull Probability Distribution Function • for x  0 b = 5 b = 3 b = 1 F(x) b = 0.5

  13. Derived from double logarithmic transformation of • the Weibull Distribution Function. • Of the form • where • Any straight line on Weibull Probability paper is a Weibull • Probability Distribution Function with slope,  and intercept, • - ln , where the ordinate is ln{ln(1/[1-F(t)])} the abscissa is • ln t. Weibull Probability Paper (WPP)

  14. Weibull Probability Paper (WPP) Weibull Probability Paper links http://perso.easynet.fr/~philimar/graphpapeng.htm http://www.weibull.com/GPaper/index.htm

  15. Use of Weibull Probability Paper b 8 4 3 2 1.5 1.0 0.8 0.7 0.5 99.0 95.0 90.0 80.0 70.0 50.0 40.0 30.0 20.0 10.0 5.0 4.0 3.0 2.0 1.0 0.5 Cumulative probability in percent F(x)in % 1.8 in. = b 1 in. 10 2 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 1000 x q

  16. Properties of the Weibull Distribution • 100pth Percentile • and, in particular • Mean or Expected Value • Note: See the Gamma Function Table to obtain values of (a)

  17. Properties of the Weibull Distribution • Standard Deviation of X • where

  18. Values of the Gamma Function The Gamma Function 

  19. f(x) Max f(x)=f(xmode) 0 x xmode Properties of the Weibull Distribution • Mode - The value of x for which the probability • density function is maximum • i.e.,

  20. Weibull Distribution - Example Let X = the ultimate tensile strength (ksi) at -200 degrees F of a type of steel that exhibits ‘cold brittleness’ at low temperatures. Suppose X has a Weibull distribution with parameters  = 20, and  = 100. Find: (a) P( X  105) (b) P(98  X  102) (c) the value of x such that P( X  x) = 0.10

  21. Weibull Distribution - Example Solution (a) P( X  105) = F(105; 20, 100) (b) P(98  X  102) = F(102; 20, 100) - F(98; 20, 100)

  22. (c) P( X  x) = 0.10 P( X  x) Then Weibull Distribution - Example Solution

  23. Weibull Distribution - Example The random variable X can modeled by a Weibull distribution with  = ½ and  = 1000. The spec time limit is set at x = 4000. What is the proportion of items not meeting spec?

  24. Weibull Distribution - Example The fraction of items not meeting spec is That is, all but about 13.53% of the items will not meet spec.

More Related