Random Variables isributions隨機變量及disributions

1、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

2、useful distributions for describing random processes in environmental science & mgt.example: 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 l

3、imit even if no leak? (prob measuring 30 even if no leak?)evidence and datan“natural” distribution of aldicarb:n500 readings from sites known to not be contaminated: evidence contdnbased on this distribution, we will assume these data are normally distributed with:nmean = 20 ppbnstandard deviation =

4、 4 ppbdefinitionsnrandom 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 ove

5、r discrete intervals.ndiscrete vs. continuous random variablefrequency 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.nexampl

6、es 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 concentration histogram (of data) vs. pdf n(20,4)known vs. unknown distributionsntrue distribution may not be a known distribution (e.g. distn of students heights in this classro

7、om)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 known easy to ask questions like the aldicarb question.discrete random variables1.bernoulli: 2

8、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

9、that occur in given time interval.normal 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

10、 normal (z):z=(x-m)/scontinuous 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

11、: inter-arrival times, “memoryless”. 6.c2: closely related to normal. non-negative. skewed.answernquestion: 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.integrate under n(20,4)nnormal pdf:sms2x21exp21)x(fndraw on boardouch!nisnt there another way?2 ways to answer1.ask s-plus (