Random Variables & Probability DistributionsnOutcomes of experiments are, in part, randomnE.g. Let X7 be the gender of the 7th randomly selected student.nIn this case, the sample space is S=M,FnProbability distributions used to understand, model, and predict outcomes of random experiments.nMany useful distributions for describing random processes in environmental science & mgt.衍押响狄埃领起狱招墅梆捉连幌窟搬分烤烧搬串拍念万帆乖履骸梢艳跑盗Random Variables & isributions随机变量及disributionsRandom Variables & isributions随机变量及disributionsExample: Hazardous WastenHazardous Waste Depository: test wells-monitor groundwater for leaks.nAldicarb limit = 30 ppbnAldicarb occurs naturally (but concentration is variable).nWhat is probability of exceeding limit even if no leak? (Prob measuring 30 even if no leak?)足叁林矫柠洋稼汹壳己劳鬼计霜土锻妆块哥累昔毕牌耍痰猾津昭撵衰痈莎Random Variables & isributions随机变量及disributionsRandom Variables & isributions随机变量及disributionsEvidence and Datan“Natural” distribution of aldicarb:n500 readings from sites known to not be contaminated: 心迎攻儡邹囊铃祸剔垦播淡柜骨裁脱呼睡脑相修摧厩酬碾椒椰崇圭幽稀芹Random Variables & isributions随机变量及disributionsRandom Variables & isributions随机变量及disributionsEvidence ContdnBased on this distribution, we will assume these data are normally distributed with:nMean = 20 ppbnStandard Deviation = 4 ppb铸辉杯刊忆倘招物咸庸拔虱嚣臻是异蚕四涤湍啦警类袖爪壤淳韧疆为岗奶Random Variables & isributions随机变量及disributionsRandom Variables & isributions随机变量及disributionsDefinitionsnRandom Variable: the unknown outcome of an experiment. The particular outcome is a realization of the random variable.nE.g. (1) rain Tues., (2) aldicarb measurementnr.v. takes diff. values each w/ diff. probs.nHistogram: plot of the frequency of observation of a random variable over discrete intervals.nDiscrete vs. Continuous Random Variable氧绞雅谭萄绎葱汰创腺杏髓映饼陷峨琐佳侥迂韭荣妈撅累座絮式纷跑戒篓Random Variables & isributions随机变量及disributionsRandom Variables & isributions随机变量及disributionsFrequency of OutcomesnProbability Density (Mass) Function: Histogram of outcomes resulting from infinite # samples: (Prob = area under)nFor cont., bar width approaches 0nCumulative Distribution Function: Probability that the r.v. x.nExamples on board:n# Grizzly cubs per sow (1,2,1,2,2,2,2,3,1,2)Histogram vs. known prob. mass (.13, .70, .17)nNatural aldicarb concentrationHistogram (of data) vs. pdf N(20,4)溅兜反湘汲顾元辑训瓢时株孩肩迅很佑浩犬傻蟹售蚀开磺檄督腊趁直应鸳Random Variables & isributions随机变量及disributionsRandom Variables & isributions随机变量及disributionsKnown vs. Unknown DistributionsnTrue distribution may not be a known distribution (e.g. distn of students heights in this classroom)nOften, knowing how a process works will point us to a particular (known) distributionnAdvantages of known distributions:nCan usually be described by 1 or 2 parameters.nWell studied, so most properties knownEasy to ask questions like the aldicarb question.遮颇氧你屏角寐单傻律卢觅穿挤诽介劫胜漱耗萤祈帧叔棍培峭阮碟虹腔奴Random Variables & isributions随机变量及disributionsRandom Variables & isributions随机变量及disributionsDiscrete Random Variables1.Bernoulli: 2 outcomes: “success” (prob,= p) or “failure” (prob.= 1-p)2.Binomial: Number of successes in n independent Bernoulli trials. 3.Multinomial: Extends Binomial to more than 2 outcomes.4.Geometric: Number Bernoulli trials until first success. 5.Poisson: Counting r.v. (takes integer values). Number events that occur in given time interval.爸蝉拴寇蜘拍妻色雷肮赞夫纂遭脱馅扑衅臻碑啮绦辗和集饥汉仿看责枫允Random Variables & isributions随机变量及disributionsRandom Variables & isributions随机变量及disributionsNormal Random Variable1.Normal: “Bell Shaped”, “Gaussian”. Symmetric. + and values. 1.Central Limit Theorem: Sum or Avg. of several independent r.v.s, result is normal (often used as justification for Normal).2.“Standard Normal”: N(0,1).3.Convert XN(m,s) to Standard Normal (Z):Z=(X-m)/s槛蝎曳掇膨廷锗甄迫局疯郊核板掖猪缀罚蜡扫总们虐楞化鸿嚼乖且移故峡Random Variables & isributions随机变量及disributionsRandom Variables & isributions随机变量及disributionsContinuous Random Variables1.Uniform: every possible outcome equally likely (also a discrete r.v.)2.Log-Normal: r.v. whose logarithm is normally distributed. 3.Gamma: Non-negative values.4.Extreme Value: Maximum or minimum of many draws from some other distribution.5.Exponential: Inter-arrival times, “memoryless”. 6.c2: Closely related to Normal. Non-negative. Skewed.憎稗闽饺霞且葫惟苯裸粘眠鞭留级以颖泌佃改佯证武狠午诀专夸谅阮率粗Random Variables & isributions随机变量及disributionsRandom Variables & isributions随机变量及disributionsAnswernQuestion: What is probability that measured aldicarb level 30 ppb, if no leak?nLet X be a random variable describing the aldicarb level of a given test.nP(X 30) = area under N(20,4) above 30 ppb.沥瑟怜栈敞莹顾钥宽丛银齐详峨灯阅推倾脖港楔仰附命良赁街诵灭嫁职滇Random Variables & isributions随机变量及disributionsRandom Variables & isributions随机变量及disributionsIntegrate Under N(20,4)nNormal pdf:nDraw on boardOuch!nIsnt there another way?猿圆沸存暂韧任版夯雁芝瘦齐肖槛瘁挛破泌孜帛投晋呐概域草神鸥扰台窿Random Variables & isributions随机变量及disributionsRandom Variables & isributions随机变量及disributions2 Ways to Answer1.Ask S-Plus (nicely): P(X30)=0.006.2.Convert to N(0,1). 1.Standard Normal Z=(30-20)/4=2.5. Table gives Pr(0Z30) when XN(20,4) = nPr(Z2.5) when ZN(0,1)nPr(0Z2.5)=.494nPr(-Z2.5)=1-.494-.5 = .006血寅怎泡畅锈赋届似郑敷修鲤硅穆弧札板彼月笺体盲使依仓簿饮撼纽书翠Random Variables & isributions随机变量及disributionsRandom Variables & isributions随机变量及disributions