2015-2016 Jan Hanning chapter-5-discrete-probability-distributions
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STATS 205
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Statistics
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Dec 18, 2024
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4
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Chapter 5 Discrete Probability DistributionsIntroduction To Statistics (University of North Carolina at Chapel Hill)Scan to open on StudocuStudocu is not sponsored or endorsed by any college or universityChapter 5 Discrete Probability DistributionsIntroduction To Statistics (University of North Carolina at Chapel Hill)Scan to open on StudocuStudocu is not sponsored or endorsed by any college or universityDownloaded by RC DAI (rcd2024@126.com)lOMoARcPSD|50751978
Statistics – Chapter 5: Discrete ProbabilityDistributions5.1 & & 5.2 Basic Concepts of ProbabilityRandom variable – a variable that has a single numerical value, determined by chance, for each outcome of a procedureProbability distribution – a description that gives the probability for each value of the random variable Discrete random variable – collection of values that is finite or countable, if thereare infinitely many values, the number of values is countable if it is possible to count them individuallyContinuous random variable – infinitely many values, and the collection of values is not countableProbability Distribution: Requirements1.There is a numerical random variable xand its values are associated with corresponding probabilities2.∑P(x) = 1 where x assumes all possible values3.0 ≤ P(x) ≤ 1 for every individual value of the random variable xProbability Distribution: GraphSimilar to a relative frequency histogram, but the vertical scale shows probabilities instead of relative frequenciesParameters of a Probability Distribution: Mean, Variance, and Standard Deviation1.Formula for Meanμ = ∑ [x×P(x)]2.Formula for Variance (easier to understand)σ2= ∑ [(x - μ)2×P(x)]3.Formula for Variance (easier for computation)σ2= ∑ [(x2×P(x)] - μ24.Standard Deviationσ = ¿−µ2∑¿√¿5.Round-off Rule for round results by carrying one more decimal place than the number of decimalplaces used for the random variable x. If the values of xare integers, roundµ, σ, or σ2 to one decimal placeExpected Value – the mean values of the outcomes, denoted by E— so E=µ.Downloaded by RC DAI (rcd2024@126.com)lOMoARcPSD|50751978
Statistics – Chapter 5: Discrete ProbabilityDistributionsMaking Sense of Results: Identifying Unusual Values1.With the Range Rule of Thumba.max .usual value=μ+2σmin.ususal value=μ−2σb.Value unusual if it is more than 2 standard deviations outside the mean2.With the Rare Event Rule for Inferential Statisticsa.Unusually highnumber of successes: x successes among ntrials is an unusually high number of successes if the probability of xor more successes is unlikely with a probability of 0.05 or less. i.P(xor more) ≤ 0.05b.Unusually lownumber of successes: x successes among ntrials is an unusually low number of successes if the probability of xor fewer successes is unlikely with a probability of 0.05 or less.i.P(xor more) ≤ 0.05Expected Value in Decision Theory Find the expected value by computing ∑ [x×P(x)]5.3 Binomial Probability DistributionsBinomial Probability Distributions Requirements1.The produce has a fixed number of trials2.The trials must be independent (the outcome of any individual trial doesn’t affect the probabilities in the other trials)3.Each trial must have 2 possible outcomes (success and failure)4.The probability of success remains the same in all trialsIndependence RequirementIf calculations are cumbersome and if the sample size is no more than 5% of the size of the population, treat the selection as being independentNotation for Binomial Probability DistributionP(S)=p→(p=probabiltyof a sucess)P(F)=q→(q=probabiltyof a failure)n=¿number of trialsx=denotesa specfifcnumberof successes∈ntrialP(x)=probabiltyof gettingexactky x successes amongthe ntrialsBinomial Probability FormulaDownloaded by RC DAI (rcd2024@126.com)lOMoARcPSD|50751978
Statistics – Chapter 5: Discrete ProbabilityDistributionsP(x)=n!(n−x)! x !× px×qn−x5.4 Parameters for Binomial ProbabilityFor Binomial DistributionsMean:μ=npVariance:σ2=npqStandard Deviation:σ=√npqRange Rule of ThumbMinimumusual value:μ−2σMaximumusual value:μ+2σDownloaded by RC DAI (rcd2024@126.com)lOMoARcPSD|50751978