Question:Why is¸Xcalled the measure of central tendency?SupposeX1; X2; :::Xn;follow the same distribution asX;and these random variables aremutually independent. Then the so-called law of large numbers (LLNs) impliesn±1nXi=1Xi!¸Xin certain sense ifn! 1. See Chapter 6 later.33
(ii)It is said that the expectation¸Xexists for a continuous distribution if and only ifZ1±1jxjfX(x)dx <1:WheneverXis a bounded random variable, that is, whenever there are numbersaandb(²1< a < b <1)such thatPr (a³X³b) = 1, then¸Xmust exist. For example, recall thep.d.f. of the Cauchy distribution isfX(x) =1·11 +x2;for² 1< x <1;thenZ1±1jxjfX(x)dx=2·Z10x1 +x2dx=1:Therefore, the expectation doesnµt exist for the Cauchy distribution.(iii)IfXis the stock return,¸Xis the expected stock return or long-run average stock return.(iv) The terminology of expectation has its origin in games of chance. This can be illustratedas follows. Four small similar chips, numbered 1, 1, 1, and 2, respectively, are placed in a bowland are mixed. A player is blindfolded and is to draw a chip from the bowl. If she draws one ofthe three chips numbered 1, she will receive one dollar. If she draws the chip numbered 2, shewill receive two dollars. It seems reasonable to assume that the player has a ²34claim³on the $1and a "14claim" on the $2. Her "total claim" is 1·34+ 2·14=54= $1:25:Thus the expectationofXis precisely the playerµs claim in this game.Theorem:IfY=aX+b;then¸Y=a¸X+b:Remark:The expectationE(±)is a linear operator.Here,a:scale parameter,b:locationparameter.Theorem:SupposeE(X2)exists. Then¸X= arg minaE(X²a)2:Proof:dE(X²a)2da= 0:thendhR1±1(x²a)2f(x)dxida=²2Z1±1(x²a)f(x)dx= 0so we havea=R1±1xf(x)dxR1±1f(x)dx=¸X34
Questions: DoesX=¸Xhas the largest probability to occur? IsP(X=¸X)the largest?Answer:No. e.g., Bernoulli r.v.:P(X=¸= 0:5) = 0gives a minimum probability.Case II:g(X) = (X²¸X)2:De°nition [Variance ofX]:of a r.v. X°2X=E(X²¸X)2=° Px(x²¸X)2fX(x);d.r.v.R1±1(x²¸X)2fX(x)dx;c.r.v.where the summation is over all possiblex0s:The standard deviation ofXis given by°X=q°2X:Remarks:(i)°2Xis a measure of the degree of spread of a distribution around its mean. A larger valueof°2Xmeans thatXis more variable. It is a scale parameter for the distribution ofX.(ii) In economics, it is interpreted as a measure of uncertainty. It is often called a measureof ²volatility³ofX.A larger value of°2Xmeans thatXis more variable.In contrast, at theextreme, if°2X= 0;thenX=¸Xwith probability 1 and there is no variation inX:Consider the d.r.v. case:°2X=Xx(x²¸X)2fX(x)=0if and only if(x²¸X)2fX(x) = 0for allx;which impliesx=¸X(no uncertainty)This is an example of degenerate distributions.