Slide 1Slide 2Slide 3Slide 4Slide 5Example 1:Example 2:To summarize…Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19CHAPTER 4• 4.1 - Discrete Models General distributions Classical: Binomial, Poisson, etc.• 4.2 - Continuous Models General distributions Classical: Normal, etc.What is the connection between probability and random variables? Events (and their corresponding probabilities) that involve experimental measurements can be described by random variables (e.g., “X = # Males” in previous gender equity example). 2POPULATIONrandom variable X3x1x2x3 x4 x5 x6 …etc….xn Data values xiRelative Frequenciesf (xi ) = fi /nx1f (x1) x2f (x2) x3f (x3) ⋮ ⋮xkf (xk)Total 1Pop values xiProbabilitiesf (xi )x1f (x1) x2f (x2) x3f (x3) ⋮ ⋮Total 1SAMPLE of size nExample: X = Cholesterol level (mg/dL)DiscretePOPULATIONPop values xProbabilitiesf (x)x1f (x1) x2f (x2) x3f (x3) ⋮ ⋮Total 1Example: X = Cholesterol level (mg/dL)random variable XDiscreteProbability HistogramXTotal Area = 1f(x) = “probability mass function” (pmf)Hey!!! What about the population mean  and the population variance  2 ???Calculating probabilities…P(a  X  b) = ????????f (x)ba�|a|x|b5POPULATIONPop values xProbabilitiesf (x)x1f (x1) x2f (x2) x3f (x3) ⋮ ⋮Total 1•Population meanAlso denoted by E[X], the “expected value” of the variable X.• Population variance  )(xfx )()( xfx22Example: X = Cholesterol level (mg/dL)random variable XDiscreteJust as the sample mean and sample variance s2 were used to characterize “measure of center” and “measure of spread” of a dataset, we can now define the “true” population mean  and population variance  2, using probabilities.xPop values xProbabilitiesf (x)x1f (x1) x2f (x2) x3f (x3) ⋮ ⋮Total 1 6POPULATION•Population meanAlso denoted by E[X], the “expected value” of the variable X.• Population variance  )(xfx )()( xfx22Example: X = Cholesterol level (mg/dL)random variable XDiscreteJust as the sample mean and sample variance s2 were used to characterize “measure of center” and “measure of spread” of a dataset, we can now define the “true” population mean  and population variance  2, using probabilities.x )/()()/()()/()( 212031106140222Pop values xiProbabilitiesf (xi )210 1/6240 1/3270 1/2 Total 1Example 1: 7POPULATION )(xfx )()( xfx22 )/)(()/)(()/)(( 2127031240612102505001/61/31/2Example: X = Cholesterol level (mg/dL)random variable XDiscrete )/()()/()()/()( 31303103130222Example 2: 8POPULATION )/)(()/)(()/)(( 312403121031180210600 )(xfx )()( xfx22Pop values xiProbabilitiesf (xi )180 1/3210 1/3240 1/3 Total 11/3 1/3 1/3Equally likely outcomes result in a “uniform distribution.”(clear from symmetry)Example: X = Cholesterol level (mg/dL)random variable XDiscreteTo summarize…910POPULATIONSAMPLE of size nx1x2x3 x4 x5 x6 …etc….xn Data xiRelative Frequenciesf (xi ) = fi /nx1f (x1) x2f (x2) x3f (x3) ⋮ ⋮xkf (xk)1Pop xiProbabilitiesf (xi )x1f (x1) x2f (x2) x3f (x3) ⋮ ⋮1Frequency TableProbability Table)()()(xfxxfx22Probability HistogramXTotal Area = 1Density HistogramXTotal Area = 1)()()(xfxxsxfxxnn221Discrete random variable X11POPULATIONSAMPLE of size nx1x2x3 x4 x5 x6 …etc….xn Data xiRelative Frequenciesf (xi ) = fi /nx1f (x1) x2f (x2) x3f (x3) ⋮ ⋮xkf (xk)1Pop xiProbabilitiesf (xi )x1f (x1) x2f (x2) x3f (x3) ⋮ ⋮1Frequency TableProbability Table)()()(xfxxfx22Probability HistogramXTotal Area = 1Density HistogramXTotal Area = 1)()()(xfxxsxfxxnn221?Discrete random variable XContinuous1213Example 3: TWO INDEPENDENT POPULATIONSX1 = Cholesterol level (mg/dL)xf1(x)210 1/6240 1/3270 1/2 Total 1X2 = Cholesterol level (mg/dL)xf2(x)180 1/3210 1/3240 1/3 Total 11 = 25012 = 5002 = 21022 = 600D = X1 – X2 ~ ???d Outcomes-30 (210, 240) 0 (210, 210), (240, 240)+30 (210, 180), (240, 210), (270, 240)+60 (240, 180), (270, 210)+90 (270, 180)NOTE: By definition, this is the sample space of the experiment!NOTE: By definition, this is the sample space of the experiment!NOTE: By definition, this is the sample space of the experiment!What are the probabilities of the corresponding events “D = d” for d = -30, 0, 30, 60, 90?NOTE: By definition, this is the sample space of the experiment!What are the probabilities of the corresponding events “D = d” for d = -30, 0, 30, 60, 90?d Outcomes-30 (210, 240) 0 (210, 210), (240, 240)+30 (210, 180), (240, 210), (270, 240)+60 (240, 180), (270, 210)+90 (270, 180)d Probabilities f(d)-30 1/9 ? 0 2/9 ?+30 3/9 ?+60 2/9 ?+90 1/9 ?14Example 3: TWO INDEPENDENT POPULATIONSX1 = Cholesterol level (mg/dL)xf1(x)210 1/6240 1/3270 1/2 Total 1X2 = Cholesterol level (mg/dL)xf2(x)180 1/3210 1/3240 1/3 Total 11 = 25012 = 5002 = 21022 = 600D = X1 – X2 ~ ???NO!!!The outcomes of D are NOT EQUALLY LIKELY!!!d Outcomes-30 (210, 240) 0 (210, 210), (240, 240)+30 (210, 180), (240, 210), (270, 240)+60 (240, 180), (270, 210)+90 (270, 180)d Probabilities f(d)-30 (1/6)(1/3) = 1/18 via independence 0 (210, 210), (240, 240)+30 (210, 180), (240, 210), (270, 240)+60 (240, 180), (270, 210)+90 (270, 180)15Example 3: TWO INDEPENDENT POPULATIONSX1 = Cholesterol level (mg/dL)xf1(x)210 1/6240 1/3270 1/2 Total 1X2 = Cholesterol level (mg/dL)xf2(x)180 1/3210 1/3240 1/3 Total 11 = 25012 = 5002 = 21022 = 600D = X1 – X2 ~ ???d Probabilities f(d)-30 (1/6)(1/3) = 1/18 via independence 0 (210, 210), (240, 240)+30 (210, 180), (240, 210), (270, 240)+60 (240, 180), (270, 210)+90 (270, 180)d Probabilities f(d)-30 (1/6)(1/3) = 1/18 via independence 0 (1/6)(1/3) + (1/3)(1/3) = 3/18+30 (210, 180), (240, 210), (270, 240)+60 (240, 180), (270, 210)+90 (270, 180)16Example 3: TWO INDEPENDENT POPULATIONSX1 = Cholesterol level (mg/dL)xf1(x)210 1/6240 1/3270 1/2 Total 1X2 = Cholesterol level (mg/dL)xf2(x)180 1/3210 1/3240 1/3 Total 11 = 25012 = 5002 = 21022 = 600D = X1 – X2 ~ ???d Probabilities f(d)-30 (1/6)(1/3) = 1/18 via independence 0 (1/6)(1/3) + (1/3)(1/3) = 3/18+30 (210, 180), (240, 210), (270, 240)+60 (240, 180),