Chapter 4 Random Variables and Probability The Citadel4章随机变量与概率的堡垒

1、chapter 5: continuous random variablesmcclave: statistics, 11th ed. chapter 5: continuous random variables2where weve beennusing probability rules to find the probability of discrete eventsnexamined probability models for discrete random variablesmcclave: statistics, 11th ed. chapter 5: continuous r

2、andom variables3where were goingndevelop the notion of a probability distribution for a continuous random variablenexamine several important continuous random variables and their probability modelsnintroduce the normal probability distribution5.1: continuous probability distributionsna continuous ra

3、ndom variable can assume any numerical value within some interval or intervals.nthe graph of the probability distribution is a smooth curve called aprobability density function,frequency function orprobability distribution.4mcclave: statistics, 11th ed. chapter 5: continuous random variables5.1: con

4、tinuous probability distributionsnthere are an infinite number of possible outcomes p(x) = 0instead, find p(axb) table software integral calculus)5mcclave: statistics, 11th ed. chapter 5: continuous random variables5.2: the uniform distributionnx can take on any value between c and d with equal prob

5、ability= 1/(d - c)nfor two values a and bdbaccdabbxap)(6mcclave: statistics, 11th ed. chapter 5: continuous random variables5.2: the uniform distributionmean:standard deviation:122cddc7mcclave: statistics, 11th ed. chapter 5: continuous random variables5.2: the uniform distributionsuppose a random v

6、ariable x is distributed uniformly with c = 5 and d = 25.what is p(10 x 18)?mcclave: statistics, 11th ed. chapter 5: continuous random variables85.2: the uniform distributionsuppose a random variable x is distributed uniformly with c = 5 and d = 25.what is p(10 x 18)?mcclave: statistics, 11th ed. ch

7、apter 5: continuous random variables940.5251018)1810( xpnthe probability density function f(x): = the mean of x = the standard deviation of x = 3.1416e = 2.71828 5.3: the normal distributionnclosely approximates many situationsperfectly symmetrical around its mean2/ )(221)(xexf10mcclave: statistics,

8、 11th ed. chapter 5: continuous random variables5.3: the normal distribution2221)(zezfneach combination of and produces a unique normal curventhe standard normal curve is used in practice, based on the standard normal random variable z ( = 0, = 1), with the probability distributionthe probabilities

9、for z are given in table iv11mcclave: statistics, 11th ed. chapter 5: continuous random variables5.3: the normal distribution0531.3413.3944.)00. 10()25. 10()25. 11 (6826.3413.3413.) 11(3413.)000. 1(3413.)00. 10(zpzpzpzpzpzp12mcclave: statistics, 11th ed. chapter 5: continuous random variables5.3: th

10、e normal distributionfor a normally distributed random variable x, if we know and , iixz13mcclave: statistics, 11th ed. chapter 5: continuous random variablesso any normally distributed variable can be analyzed with this single distribution5.3: the normal distributionnsay a toy car goes an average o

11、f 3,000 yards between recharges, with a standard deviation of 50 yards (i.e., = 3,000 and = 50) nwhat is the probability that the car will go more than 3,100 yards without recharging?14mcclave: statistics, 11th ed. chapter 5: continuous random variables5.3: the normal distribution0228.4772.5 .1)00.

12、20(5 .1)00. 2(1)00. 2(5030003100)3100(zpzpzpzpxpnsay a toy car goes an average of 3,000 yards between recharges, with a standard deviation of 50 yards (i.e., = 3,000 and = 50) nwhat is the probability that the car will go more than 3,100 yards without recharging?15mcclave: statistics, 11th ed. chapt

13、er 5: continuous random variables5.3: the normal distributionnto find the probability for a normal random variable sketch the normal distributionindicate xs meanconvert the x variables into z valuesput both sets of values on the sketch, z below xuse table iv to find the desired probabilities16mcclav

14、e: statistics, 11th ed. chapter 5: continuous random variables5.4: descriptive methods for assessing normalitynif the data are normala histogram or stem-and-leaf display will look like the normal curvethe mean s, 2s and 3s will approximate the empirical rule percentagesthe ratio of the interquartile

15、 range to the standard deviation will be about 1.3a normal probability plot , a scatterplot with the ranked data on one axis and the expected z-scores from a standard normal distribution on the other axis, will produce close to a straight line17mcclave: statistics, 11th ed. chapter 5: continuous ran

16、dom variables5.4: descriptive methods for assessing normalityerrors per mlb team in 2003mean: 106standard deviation: 17iqr: 2229. 11722siqr15755511063140723410621238917106sxsxsx22 out of 30: 73%28 out of 30: 93%30 out of 30: 100% 18mcclave: statistics, 11th ed. chapter 5: continuous random variables

17、5.4: descriptive methods for assessing normalitya normal probability plot is a scatterplot with the ranked data on one axis and the expected z-scores from a standard normal distribution on the other axis 19mcclave: statistics, 11th ed. chapter 5: continuous random variables5.5: approximating a binom

18、ial distribution with the normal distributionndiscrete calculations may become very cumbersomenthe normal distribution may be used to approximate discrete distributionsthe larger n is, and the closer p is to .5, the better the approximationnsince we need a range, not a value, the correction for cont

19、inuity must be useda number r becomes r+.520mcclave: statistics, 11th ed. chapter 5: continuous random variablescalculate the mean plus/minus 3 standard deviationsnpqnp3if this interval is in the range 0 to n, the approximation will be reasonably closeexpress the binomial probability as a range of v

20、alues)()()(axpbxpaxpfind the z-values for each binomial value)5 .(azuse the standard normal distribution to findthe probability for the range of values you calculated21mcclave: statistics, 11th ed. chapter 5: continuous random variables5.5: approximating a binomial distribution with the normal distr

21、ibutionflip a coin 100 times and compare the binomial and normal results0796.)10. 010. 0(5505 .505505 .49)5 .505 .49(55 .5 .100505 .1000796.5 .5 .50100)50(5050zpzpxpxp22mcclave: statistics, 11th ed. chapter 5: continuous random variablesbinomial:normal:5.5: approximating a binomial distribution with

22、 the normal distributionflip a weighted coin p(h)=.4 10 times and compare the results1255.)32. 032. 0(55. 145 . 555. 145 . 4)5 . 55 . 4(55. 16 .4 .1044 .101204.6 .4 .510)5(55zpzpxpxp23mcclave: statistics, 11th ed. chapter 5: continuous random variablesbinomial:normal:5.5: approximating a binomial di

23、stribution with the normal distributionflip a weighted coin p(h)=.4 10 times and compare the results1255.)32. 032. 0(55. 145 . 555. 145 . 4)5 . 55 . 4(55. 16 .4 .1044 .101204.6 .4 .510)5(55zpzpxpxp24mcclave: statistics, 11th ed. chapter 5: continuous random variablesbinomial:normal:5.5: approximating a binomial distribution with the normal distributionthe more p differs from .5, and the smaller n is,the less precise the approximation will be5.6: the exponential distribution25mcclave: statistics, 11th ed. chapter 5: continuous random variablesnprobability distribution