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Continuous Random Variables

Continuous Random Variables. Continuous Random Variables Infinite Number of Possibilities. Discussion topics Cumulative distribution functions Method of calculation Relationship to pdf General characteristics of a continuous rv Mean and variance Standard models

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Continuous Random Variables

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  1. Continuous Random Variables statistical processes

  2. Continuous Random Variables Infinite Number of Possibilities • Discussion topics • Cumulative distribution functions • Method of calculation • Relationship to pdf • General characteristics of a continuous rv • Mean and variance • Standard models • Use as models for physical processes • Testing for normality statistical processes

  3. F(y0) y0 Cumulative Distribution Functions What does the CDF look like for a discrete random variable?? F(y0) = p(y  y0) statistical processes

  4. F(y1) F(y0) y0 y1 CDFs - Calculating p(y0  y y1) p(y0  y  y1) = F(y1) - F(y0) statistical processes

  5. CDFs of Continuous RVs - Some Characteristics • CDF is continuous • What about for a discrete rv • Monotonically increasing function • Has asymptotes at 0 and 1 • Does F(y) have to be 0 & 1 at the extremes? • Some previously encountered values F(m) = 0.50  median F(QL) = 0.25  lower quartile F(QU) = 0.75  upper quartile statistical processes

  6. Relationships Involving CDF & PDF For a continuous rv, what is p(y = y0)? statistical processes

  7. Function g of y Pdf of y Expected Values - Continuous RVs Recall that E( ) is a linear operator. statistical processes

  8. May be 1, 2, … “about the point y0” Moments - Useful Descriptors of Distributions • General form for a moment of a rv E(y - y0)k Above read as: “the kth moment about the point y0” • What is the 1st moment about the origin? • What is the 2nd moment about the mean? statistical processes

  9. Moments – Lower Order Moments • 1st moment about the origin E(y - 0)1 = E(y) • 2nd moment about the mean E(y - μ)2 = VAR(y) statistical processes

  10. Higher Order Moments - About the Mean • Skewness is 3rd moment about the mean E[(y - )3] • If > 0  skewed to the right • If < 0  skewed to the left • If = 0 symmetric • Kurtosis is 4th moment about the mean E[(y - )4] • If > 0  fat tailed • If < 0  thin tailed • If = 0 normal distribution • Look at results from Excel Descriptive Statistics statistical processes

  11. Kurtosis negative – thin tailed Positive skewness – to the right Higher Order Moments – About the Mean statistical processes

  12. Continuous Distribution FunctionsSome Commonly Encountered Models • Recall purpose of describing standard distribution functions • Models for processes found in nature • Building blocks for statistical analysis • Many, many other standard models • Specialized purposes statistical processes

  13. Why does this have to be this value? Uniform Distribution- A Simple Model f(y) 1/(b-a) y a b When would you use this type of distribution function? Can you think of an application for this function? What does the CDF look like? statistical processes

  14. Uniform Distribution Function statistical processes

  15. c = 14 b = 6 a = 2 Triangular Distribution What do the parameters, a, b & c represent? When would this be a useful model? statistical processes

  16. Triangular Distribution – PDF & CDF for y~Triang(a,b,c) statistical processes

  17. Triangular Distribution - Mean & Variance statistical processes

  18. Note convention  = 5 Location parameter Normal Distribution - A Bell Curve 2 is shape parameter Would you use the above to model the weight of an object? What is a truncated normal distribution? statistical processes

  19. y~N(, 2) PDF z the standard normal variable z = (y - ) /  y ~ N(, 2) z ~ N(0, 1) What does this do? Note why if y is a rv then z must be a rv. Normal Distributions - Characteristics & Extension • CDF • No closed form expression • What does this mean & imply? statistical processes

  20. Transforming to z Values Why are transformations to z values useful? 1) Detection of outliers easier 2) F(y0) = F(z0) • What does the above mean? • Can produce tables of F(z) • “Z” Table in back of Text statistical processes

  21. For a given value z0, Table 4 gives the area A (shown above) Using Standard Normal Tables z0 Let z0 = 0.27, what is A? Verify the empirical rule. What is interpretation of A? statistical processes

  22. Finding Probabilities - y~N(, 2) Given y~N(120, 152), find • p(y  120) • p(y < 120) • p(y < 108) • p(115  y  124) statistical processes

  23. statistical processes

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  25. statistical processes

  26. SW use these: Best Fit Stat:Fit many others Assessing Normality • Objective • Given a set of observations of rv y • Is y a normally distributed rv? What is a more general problem? • Methods of addressing more general problem • 2 test (read as Chi squared) • Kolmogorov-Smirnov test • Anderson-Darling test Why would you be interested in knowing? statistical processes

  27. Normality Tests - Visual Inspection • Visual inspection • Construct plot (histogram, stem & leaf, etc.) • Does it look normal? • Advantages of method • Simple, no special expertise needed to conduct or interpret • Disadvantages • Crude, no specific measures • Dependent on analyst statistical processes

  28. Normality Tests - Interquartile Range • Calculate IQR • Recall IQR? • Find ratio IQR/s • s is standard deviation of sample observations • If IQR/s  1.3  sample is ~N • Why should IQR/s  1.3? • For z  (QU - QL) / s = [0.67 - (-0.67)] / 1 = 1.34 What are advantages & disadvantages? statistical processes

  29. Normality Tests - Normal Probability Plot • Several approaches • Same general concept • Plot observations vs. E(observation if ~N) • Should be approximately linear relationship • Steps • Generate n random samples of zi from ~ N(0,1) • Convert zi to corresponding yi • Generating E(observation if ~N) • Rank all yi • Plot the actual observations vs. randomly generated yi What should the plot look like? And why?? statistical processes

  30. Continuing With Standard Models - The Gamma Distribution y~ ( , ) • PDF • CDF • No closed form solution • Interesting characteristics • Only assumes non-negative values • Described by two parameters •  a shape parameter •  a scale parameter  =   ; 2 =   2 statistical processes

  31. Gamma becomes a 2 distribution when  = /2 and  = 2 (for  an integer), y~X2() Turns out even more interesting The 2 DistributionA Special Case of the Gamma statistical processes

  32. Preview - Coming Attractions • Uses of the 2 distribution • Estimating the variance of a population • Goodness of fit test • Take sample observations • Postulate distribution of parent population • How good is fit of postulated distribution? • Will discuss both of above during course statistical processes

  33. Gamma becomes an exponential when  = 1 and  = any constant, y~Exp( ) Interesting characteristic Memoryless distribution (aka Markov property) Exponential DistributionAnother Special Case of a Gamma statistical processes

  34. s t What Do We Mean byMemoryless What does this mean in English? statistical processes

  35. Exponential & Poisson - Two Sides to the Same Coin What is y for each of the above? What are  and , respectively? What is the relationship between  and ? Can you see relationship between exp. & poisson? statistical processes

  36. Exponential & Poisson - A More Formal Relationship • Consider case of incoming phone calls x # of calls in 1 minute, x ~ Poi() • If just received call, what is p(>30 minutes to next call)? • Let y  # of calls in t minutes y ~ Poi(t)  Why?? statistical processes

  37. Continuing With Example • What happens when you calculate p(y=0)? If above is p(T > t), what is p(T  t)?? What does p(T  t) remind you of?? statistical processes

  38. Seeing Relationships - Exponential & Poisson statistical processes

  39. Seeing Relationships - Exponential & Poisson The same expressions! RV y is general expression statistical processes

  40. Exponential Distribution - Typical Form statistical processes

  41. Exercise • Consider following problem • You manufacture power supplies that exhibit on average 2 failures every 8000 hours of operation • What is p(you see 2 failures in 4000 hours of operation)? • What is p(power supply will operate >6000 hours without a failure)? statistical processes

  42. Almost like saying 1 failure in 4000 hours Exercise statistical processes

  43. Exercise (continued) Can you solve this using the Poisson distribution?? statistical processes

  44. Weibull Distribution y~W( , ) • Valid only for y  0 Comments on Weibull • Also used for reliability modeling • May also degenerate into exponential distribution statistical processes

  45. Beta Distribution y~ B( , ) • Valid for [0,1] Comments on Beta • What physical processes can we model that fall in [0,1]? • How can we make outside [0,1]? • Useful for project scheduling statistical processes

  46. Class 4 Readings & Problems • Reading assignment • M & S • Chapter 5 Sections 5.1 - 5.9 • B & C • Chapter 5 • Recommended problems • M & S Chapter 5 • 62, 82, 86, 87,88,89, 96, 98 statistical processes

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