Slide 1Example: Continuous r.v.Development of pdfpdfExample 1: pdf UniformExample 2: pdfpdf/cdfExample cdf: UniformF(x): UniformF(x): Uniform (general case)Example cdf: Uniform (cont)Example: PercentileRules of Expected ValuesExample: ExpectationsVarianceExample: ExpectationsNormal DistributionShapes of Normal CurvesShape of z curve(z)Slide 22Slide 23Using the Z tableSymmetry of z-curvezExample: Nonstandard Normal DistributionEmpirical RuleExample: Nonstandard Normal DistributionContinuity CorrectionContinuity Correction - ProcedureExample: Approximating a BinomialShape of ExponentialExample: Exponential DistributionGamma Distribution: usesGamma FunctionGamma DistributionShapes of Gamma DistributionGamma Distribution2 distributionShapes of χ2 DistributionWeibull – pdfWeibull – UsesWeibull Distribution: ShapesWeibull – Expectation/VarianceWeibull – cdfLognormal – UsesLognormal – pdfLognormal Distribution: ShapesLognormal – Expectation/VarianceLognormal – cdfBeta – usesBeta – pdfBeta Distribution:Shapes (Standard)Beta – Expectation/VarianceQQ Plot: PercentilesQQ-plot - normalQQ-plot – light tailsQQ-plot: heavy tailedQQ-plot: right skewedQQ Plot – Left SkewedCopyright © Cengage Learning. All rights reserved. 4Continuous Random Variables and Probability Distributionshttps://onlinecourses.science.psu.edu/stat414/node/307Example: Continuous r.v.In a computer repair shop, select computers that are brought in at random. Let X = the time that a computer functions before breaking down.Select runners at random in a certain park. Let X = the distance run between seeing two people while running in the park.Make depth measurements at a randomly selected location in a specific lake. Let X = the depth at this location.A chemical compound is randomly selected. Let X = the pH value of the compound measured in a solvent.Development of pdf(a)(b)(c)pdfP(a  X  b)Example 1: pdf UniformA person casually walks to the bus stop when the bus comes every 30 minutes. What is the pdf for the wait time?What is the probability that the person has to wait between 5 and 10 minutes?What is the probability that the person has to wait longer than 5 minutes?Example 2: pdfLet X = the life span of some bacteria (in hours), X is a continuous r.v. with pdfWhat is the probability that the bacteria lives over 2 hours?What is the probability that the bacteria dies within one hour?2x2e x 0f(x)0 else-��=��pdf/cdfA pdf and associated cdfhttp://daad.wb.tu-harburg.de/?id=271Example cdf: UniformA person casually walks to the bus stop when the bus comes every 30 minutes has a pdf ofWhat is the cdf of X? 10 x 30f(x)300 else�� ��=���F(x): Uniform-10 0 10 20 30 40 5001F(x): Uniform (general case)01ABExample cdf: Uniform (cont)A person casually walks to the bus stop when the bus comes every 30 minutes. Use F(x) to make the following calculations.What is the probability that the person has to wait between 5 and 10 minutes?What is the probability that the person has to wait longer than 5 minutes?Example: PercentileThe distribution of the grade of a particular road in a particular 2 mile region is a continuous r.v. X with pdfWhat is the 50th percentile?•Rules of Expected Values•E(aX + b) = aE(X) + b•For r.v. X1, X2, …, XnE(a1X1 + … + anXn) = a1E(X1) + … anE(Xn)•Example: ExpectationsThe uniform distribution has a pdf ofWhat are E(X) and E(X2)? 1A x Bf(x)B A0 else�� ��=-���Variance•Var(X) = E(X2) – (E(X))2Rules: Given two real numbers a and b and a function h•Var(aX + b) = a2Var(X)•aX+b = |a|X•Var[h(X)] = E[h2(X)] – [E(h(X))]2Example: ExpectationsThe uniform distribution has a pdf ofWhat are E(X) and E(X2)? What is the Var(X)?1A x Bf(x)B A0 else�� ��=-���Normal DistributionA continuous r.v. X is said to have a normal distribution with parameters μ and σ (σ2), where - < μ <  and σ > 0, if the pdf of X is•Shapes of Normal Curveshttps://en.wikipedia.org/wiki/File:Normal_Distribution_PDF.svgShape of z curve(z)Using the Z tableSymmetry of z-curvezExample: Nonstandard Normal DistributionSuppose the current measurements in a strip of wire are assumed to follow a normal distribution with a mean of 10 mA (milliamperes) and a standard deviation of 2.1 mA. a) What is the probability that a current measurement will be between 9 mA and 13 mA?b) What is the probability that a current measurement will exceed 13 mA.Empirical Rulehttp://www.learner.org/courses/againstallodds/about/glossary.htmlExample: Nonstandard Normal DistributionSuppose the current measurements in a strip of wire are assumed to follow a normal distribution with a mean of 10 mA (milliamperes) and a standard deviation of 2.1 mA. a) What is the probability that a current measurement will be between 9 mA and 13 mA?b) What is the probability that a current measurement will exceed 13 mA.c) Determine the 95th percentile of the current measurements?Continuity Correctionhttp://faculty.cns.uni.edu/~campbell/stat/prob9.htmlContinuity Correction - ProcedureActual Value Approximate ValueP(X = a) P(a – 0.5 < X < a +0.5)P(a < X) P(a + 0.5 < X)P(a ≤ X) P(a – 0.5 < X)P(X < b) P(X < b – 0.5)P(X ≤ b) P(X < b + 0.5)Example: Approximating a Binomial72% of women marry before 35 years old. For 500 women, what is the probability that at least 375 get married before they are 35 years old?Shape of Exponentialhttp://en.wikipedia.org/wiki/File:Exponential_pdf.svgExample: Exponential DistributionThe time, in hours, during which an electrical generator is operational is a r.v. that follows the exponential distribution with expected time of operation of 160 hours. What is the probability that the generator of this type will be operational fora) less than 40 hours?b) between 60 and 160 hours?c) more than 200 hours?Gamma Distribution: uses•Interval or time to failure (Exponential)•Queuing models•Flow of items through manufacturing and distribution processes•Load on web servers•Telecom exchange•Climatology – model for rainfall•Financial services – insurance claims, size of load defaults, probability of ruin, value of riskGamma FunctionFor  > 0, Properties:1) For  > 1, () = ( – 1)  ( – 1)2) For any positive integer n, (n) = (n – 1)!3) 12� �G = p� �� �1 x0( ) x e dx�a- -G a =�Gamma DistributionStandard:  =1Exponential:  = 1,  = 1/1 x/1x e x 0( )f(x; , )0 elsea- - ba���b G aa b