1 / 76

Chapter Four Continuous Random Variables & Probability Distributions

Chapter Four Continuous Random Variables & Probability Distributions. Continuous Probability Distribution (pdf) Definition: b P(a  X  b) =  f(x) dx a For continuous RV X & a  b. Two Conditions for a pdf: 1) f(x)  0 for all x + 2)  f(x) dx = 1 - .

galvin
Télécharger la présentation

Chapter Four Continuous Random Variables & Probability Distributions

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. Chapter FourContinuous Random Variables & Probability Distributions

  2. Continuous Probability Distribution (pdf) Definition:b P(a  X b) =  f(x)dxaFor continuous RV X & a b.Two Conditions for a pdf: 1) f(x)  0 for all x+ 2)  f(x)dx= 1 -

  3. Cumulative Distribution FunctionDefinition:a F(a) = P(X  a) = f(x)dx-For continuous RV X.

  4. PDF ExampleGiven pdf:f(x) = cx2, for 0  x  2f(x) = 0, elsewhere.Find the value of c for which f(x) will be a valid pdf.Find the probability that 1  x  2.

  5. Cumulative Distribution FunctionPlot the CDF for RV X: F(x) = 0 for x < 0 F(x) = x for 0  x  1 F(x) = 1 for x > 1.

  6. Expected Values for Continuous RV+ E(X) = x =  x f(x)dx-Variance+V(X) = 2x =  (x-)2f(x)dx=E[(X-)2]-Shortcut: V(X) = E(X2) – [E(X)]2

  7. Expected Value of RV X ExampleFor a RV X, the pdf is given by:f(x) = 0 for x < 50f(x) = 1/20 for 50  x  70f(x) = 0 for x > 70. What is the expected value & variance of RV X?

  8. Expected Value & Variance ExampleAn electronic component is tested to failure. Let continuous RV X measure the time to failure for this component. The pdf for this electronic component is shown to be in hours by the following:f(t) = 0 for t < 0f(t) = e-t/1,000 for t  0. 1,000What is the expected value & variance of RV X?

  9. Continuous Probability Distributions Uniform Normal Gamma Exponential Chi-Squared Weibull Lognormal Beta

  10. Uniform Distributionf(x; a, b) = 1 a  x  bb-a

  11. Uniform pdf ExampleA telephone call is equally likely to occur any time between Noon & 1 PM. Let X be a RV that is 0 at Noon & is the time of the call in fractions of a hour past Noon for any other outcome.Find P(0  X  1.0).Find P(X = 0.2).Find P(X  0.5).Find P(0.2 < X < 0.8).

  12. Uniform Probability DistributionResistors are produced that have a nominal value of 10 ohms & are +/- 10% resistors where any possible value of resistance is equally likely.Find the pdf & cdf of the RV X, which represents resistance.Find the probability that a resistor, selected at random, is between 9.5 & 10.5 ohms.

  13. Uniform pdfThe time X in hours for an engineering student to complete a homework assignment in ISE 261 that receives a grade above 85% is uniformly distributed with a minimum effort of 3 hours & a maximum effort of 10 hours. What is the probability that preparation time exceeds 5 hours?What is the probability that the preparation time is within 2 hours of the mean time?

  14. Normal Distribution2 2f(x;,) = e-(x-)/ 2 (2)-< x <+Abbreviated: X  N(,2)

  15. Normal Distribution Intervalb 2 2 P(a<x<b) =  e-(x-) / 2 / (2) dx a

  16. Properties of Normal pdf’sBell ShapedSymmetrical about the MeanUnimodalMean = Median = ModeOne Standard Deviation is at the inflection points on the curveThe Standard Deviation defines the area under the pdf

  17. Standard Normal Distribution2f(z;0,1) = e-z/ 2 (2)-< z <+Denote: N(0,1) by Z CDF(Z) by (z)

  18. Z TransformationFor RV X with N(,2):Z = X -  has a standard Normal pdfThusP(aXb)=P[ a-   Z  b- ] 

  19. Transformation ExampleFind the probability that RV X is greater than 5 if X is N(3,4).

  20. Normal Distribution ExampleThe time to wear out of a cutting tool edge is distributed normally with  = 2.8 hours &  = .60 hour. What is the probability that the tool will wear out in less than 1.5 hour?How often should the cutting edges be replaced to keep the failure rate less than 10% of the tool’s?

  21. Normal DistributionYou are an engineering manager of a PCB operation where the mean thickness of a board should be 65mm with a SD of 2mm. Previous studies indicate that this operation can be assumed to be normally distributed.One of the boards inspected has just produced a thickness of 68mm. What is the probability that a board greater than this thickness can be produced from this process?

  22. Normal Distribution ExampleThe achievement scores for a college entrance exam are normally distributed with a mean equal to 75 & standard deviation equal to 10. What fraction of the scores are expected to lie between 70 & 90?

  23. Normal Distribution ExampleThe width of a slot on a duralumin forging is normally distributed with mean equal to 0.90 inch & SD equal to 0.002 inch. The specification limits are given as 0.90 +/- 0.005 inch. What percentage of forgings will be defective?

  24. Normal Distribution ExampleA process manufactures ball bearings whose diameters (in cm) are N(2.505, 0.0082). Specifications call for the diameter to be in the interval 2.5 +/- 0.01 cm. What proportion of the ball bearings will meet the specification?

  25. Normal DistributionA remote computer sends a 1 or 0 by sending either p(t) or –p(t), respectively… where the pulse p(t) is given by:p(t) = 1, 0  t  T 0, otherwiseWhen the pulse is received it is corrupted by Gaussian noise which is added to the signal. Assume that this noise has zero mean and variance of 1. Suppose that you wish to measure a single value within a received pulse and use that to determine if a 1 or 0 was sent. Then the value that you measure would be a Gaussian RV Z modeled as follows:Z = ± 1 + VWhere V is a Gaussian RV with zero mean and variance of 1. You decide that a 1 was sent if the measured Z is such that Z > 0 and otherwise will decide that a 0 was sent. Consider the case where a 1 was sent (thus Z = +1 +V). What is the probability of making an error?

  26. Gamma Probability DistributionFor RV X  0:f(x; , ) = x -1 e-x/ ()Where  > 0,  > 0.

  27. Gamma FunctionDefinition:() =  x-1 e-x dx0 For  > 0.

  28. Properties of the Gamma Function(1) = 1(1/2) =  For any positive integer n: (n) = (n-1)!For  >1: () = (-1) (-1)

  29. Gamma pdf PropertiesStandard Gamma pdf when  =1 (Incomplete Gamma function) E(X) =  V(X) = 2Exponential pdf when  = 1 Chi-Square pdf when  = v/2 & 

  30. Gamma TransformationP(X  x) = F(x , , ) = F(x/, )x F(x/, ) = y -1 e-y /() dy 0 x > 0

  31. Gamma pdf ExampleThe chemical reaction time (in seconds) of a commercial grade of epoxy can be represented with a standard Gamma pdf with parameter  =5.0 seconds. What is the probability that the reaction time will be more than 4 seconds?

  32. Gamma DistributionIn simple materials, all molecules are the same and have the same molecular weight. Because polymers contain molecules of different sizes, engineers must use probability distributions to express molecular weight.A commercial grade of polymer has a molecular weight with a standard Gamma distribution with  = 2 molar mass. What is the probability that your randomly selected sample has a molar mass less than 3?

  33. Gamma pdf ExampleThe arrival time of large comets into our solar system has followed a Gamma distribution with parameters  = 4 years &  = 15.What is the expected arrival time of large comets?What is the variance in arrival time?What is the probability of a visit by a large comet between the years 2024 & 2039?

  34. Gamma DistributionA computer firm introduces a new home computer. Past experience indicates that the time of peak demand (in months) after its introduction, follows a Gamma pdf with Variance = 36.0.If the expected value is 18 months, find  and .What is the probability that the peak demand is less than 7 months?

  35. Exponential Probability Distributionf(x; ) =  e-xx  0Where  > 0.

  36. Exponential pdf Properties E(X) =  = 1 /  V(X) = 2 = 1 / 2  = 1 /  pdf easily integrated:xCDF =  e-tdt =1 – e-x0

  37. Example Exponential pdfA radioactive mass emits particles according to a Poisson process at a mean rate of 15 per minute. At some point, a clock is started. What is the probability that more than 5 seconds will elapse before the next emission?What is the mean waiting time until the next particle is emitted?

  38. Exponential pdf Lack of Memory PropertyThe lifetime of an electrical component has an Exponential pdf with mean = 2 years. What is the probability that the component lasts longer than 3 years?Assume the component is now 4 years old, and is still functioning. What is the probability that it functions for more than 3 additional years?

  39. Exponential ExampleSome types of equipment have been observed to fail according to an Exponential pdf. The time to failure T for these types has a mean time to failure (MTTF) equal to 1/.The mean time to failure of a light bulb is 100 hours, what is the probability that a light bulb will last for more than 150 hours?

More Related