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1、数理统计复习北京外国语大学国际商学院应惟伟1计量经济学 数理统计复习主要内容期望 Expected value方差Variance协方差Covariance2计量经济学 数理统计复习Definition of E(X),the expected value of X:EXPECTED VALUE OF A RANDOM VARIABLEThe expected value of a random variable,also known as its population mean,is the weighted average of its possible values,the weight
2、s being the probabilities attached to the values.3计量经济学 数理统计复习The expected value turns out to be 7.Actually,this was obvious anyway.We saw in the previous sequence that the distribution is symmetrical about 7.EXPECTED VALUE OF A RANDOM VARIABLExipixi pixi pi xi pi x1p1x1 p121/362/36x2p2x2 p232/366/3
3、6x3p3x3 p343/3612/36x4p4x4 p454/3620/36x5p5x5 p565/3630/36x6p6x6 p676/3642/36x7p7x7 p785/3640/36x8p8x8 p894/3636/36x9p9x9 p9103/3630/36x10p10 x10 p10112/3622/36x11p11x11 p11121/3612/36 S S xi pi=E(X)252/36=74计量经济学 数理统计复习Alternative notation for E(X):E(X)=m mX Very often the expected value of a rando
4、m variable is represented by m m,.If there is more than one random variable,their expected values are differentiated by adding subscripts to m m.EXPECTED VALUE OF A RANDOM VARIABLE5计量经济学 数理统计复习 Expected value rulesThere are three rules that we are going to use over and over again.Rule 1 E(X+Y+Z)=E(X
5、)+E(Y)+E(Z)Rule 2 E(bX)=b E(X)where b is a constant.Rule 3 E(b)=b where b is a constant6计量经济学 数理统计复习 Definition of Eg(X),the expected value of a function of X:Example:For example,the expected value of X2 is found by calculating all its possible values,multiplying them by the corresponding probabilit
6、ies,and summing.EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE7计量经济学 数理统计复习 Population variance of X:The expected value of the squared deviation is known as the population variance of X.It is a measure of the dispersion of the distribution of X about its population mean.POPULATION VARIANCE OF A D
7、ISCRETE RANDOM VARIABLE8计量经济学 数理统计复习=E(X2)-m m2=E(X-m m)2=E(X2-2m mX+m m2)=E(X2)+E(-2m mX)+E(m m2)=E(X2)-2m mE(X)+m m2 =E(X2)-2 m m2+m m2 =E(X2)-m m2 ALTERNATIVE EXPRESSION FOR POPULATION VARIANCE9计量经济学 数理统计复习 Population variance of XIn equations,the population variance of X is usually written s sX2
8、.POPULATION VARIANCE OF A DISCRETE RANDOM VARIABLE10计量经济学 数理统计复习 Two random variables X and Y are said to beindependent if and only ifEf(X)g(Y)=Ef(X)Eg(Y)for any functions f(X)and g(Y).Special case:if X and Y are independent,E(XY)=E(X)E(Y)INDEPENDENCE OF TWO RANDOM VARIABLESAs a special case,the exp
9、ected value of XY is equal to the expected value of X multiplied by the expected value of Y if and only if X and Y are independent.11计量经济学 数理统计复习 Variance rulesRule 1 If Y=V+W,Var(Y)=Var(V)+Var(W)+2 Cov(V,W)Rule 2 If Y=bZ,where b is a constant,Var(Y)=b2 Var(Z)Rule 3 If Y=b,where b is a constant,Var(
10、Y)=0Rule 4 If Y=V+b,where b is a constant,Var(Y)=Var(V)We can get proof of these rules by definition of variance,however,we have alternative proof in the later part.12计量经济学 数理统计复习Definition of sample covariance:SAMPLE COVARIANCE:EXAMPLE CALCULATIONGiven a sample of n observations on two variables X
11、and Y,the sample covariance is the average of the products of their deviations about their sample means.13计量经济学 数理统计复习 Observation S Y11517.2421615.003814.91464.5051518.006126.2971219.2381818.699127.21102042.0619127.5020148.00 SAMPLE COVARIANCE:EXAMPLE CALCULATIONThe table above shows S,the years of
12、 schooling(highest grade completed)and Y,the hourly earnings.14计量经济学 数理统计复习SAMPLE COVARIANCE:EXAMPLE CALCULATIONWe will calculate the sample covariance of S and Y.Here the data are plotted in a scatter diagram.15计量经济学 数理统计复习 Observation S Y S-Y-(S-)(Y-)11517.241.753.0165.27721615.002.750.7762.133381
13、4.91-5.250.686-3.599464.50-7.25-9.72570.50351518.001.753.7766.6076126.29-1.25-7.9359.91871219.23-1.255.006-6.25781818.694.754.46621.2119127.21-1.25-7.0158.768102042.066.7527.836187.890.19127.50-1.25-6.7258.40620148.000.75-6.225-4.668Total265284.49305.888Average13.2514.22515.294SAMPLE COVARIANCE:EXAM
14、PLE CALCULATIONThe products are summed and divided by 20,the number of observations.The sample covariance is thus 15.29.16计量经济学 数理统计复习1.If Y=V+W,Cov(X,Y)=Cov(X,V)+Cov(X,W)2.If Y=bZ,where b is a constant,Cov(X,Y)=Cov(X,bZ)=bCov(X,Z)Example:Cov(X,3Z)=3Cov(X,Z)3.If Y=b,where b is a constant,Cov(X,Y)=Co
15、v(X,b)=0Example:Cov(X,10)=0COVARIANCE RULES17计量经济学 数理统计复习 Example:suppose Y=b1+b2ZCov(X,Y)=Cov(X,b1+b2Z)=Cov(X,b1)+Cov(X,b2Z)=0+Cov(X,b2Z)=b2Cov(X,Z)COVARIANCE RULES18计量经济学 数理统计复习1.If Y=V+W,Cov(X,Y)=Cov(X,V)+Cov(X,W)Proof of COVARIANCE RULES19计量经济学 数理统计复习Proof of COVARIANCE RULES2.If Y=bZ,where b is
16、 a constant,Cov(X,Y)=Cov(X,bZ)=bCov(X,Z)20计量经济学 数理统计复习3.If Y=b,where b is a constant,Cov(X,Y)=Cov(X,b)=0Proof of COVARIANCE RULES21计量经济学 数理统计复习Proof of VARIANCE RULESVariance Rule 1:If Y=V+W,Var(Y)=Var(V)+Var(W)+2Cov(V,W)Proof:Var(Y)=Cov(Y,Y)=Cov(Y,V+W)=Cov(Y,V)+Cov(Y,W)=Cov(V+W,V)+Cov(V+W,W)=Cov(V,
17、V)+Cov(W,V)+Cov(V,W)+Cov(W,W)=Var(V)+Var(W)+2Cov(V,W)Note that the order of the variables makes no difference when defining covariance and hence Cov(W,V)is the same as Cov(V,W).22计量经济学 数理统计复习Proof of VARIANCE RULESVariance Rule 2:If Y=bZ,where b is a constant,Var(Y)=b2Var(Z)Proof:Var(Y)=Cov(Y,Y)=Cov
18、(Y,bZ)=bCov(Y,Z)=bCov(bZ,Z)=b2Cov(Z,Z)=b2Var(Z)23计量经济学 数理统计复习Variance Rule 3:If Y=b,where b is a constant,Var(Y)=0Proof:Var(Y)=Cov(Y,Y)=Cov(b,b)=0 Proof of VARIANCE RULES24计量经济学 数理统计复习Proof of VARIANCE RULESVariance Rule 4:If Y=V+b,where b is a constant,Var(Y)=Var(V)Proof:Var(Y)=Var(V+b)=Var(V)+Var(
19、b)+2Cov(V,b)=Var(V)25计量经济学 数理统计复习 ALTERNATIVE EXPRESSION FOR SAMPLE COVARIANCEWe now sum vertically.The first summation is the summation of the first terms in the n lines.26计量经济学 数理统计复习 ALTERNATIVE EXPRESSION FOR SAMPLE COVARIANCETwo of the terms cancel and we arrive at the alternative expression.Note that the 1/n factor applies only to the first term.27计量经济学 数理统计复习POPULATION COVARIANCE If X and Y are independent,s sXY=0The expected values of both factors are zero because E(X)=m mX and E(Y)=m mY.E(m mX)=m mX and E(m mY)=m mY because m mX and m mY are constants.28计量经济学 数理统计复习