Transcription of Lecture 6: Discrete Random Variables
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Lecture 6: Discrete Random Variables
Lecture 6: Discrete Random Variables19 September 20051 ExpectationThe expectation of a Random variable is its average value, with weights in theaverage given by the probability distributionE[X] = xPr (X=x)xIfcis a constant,E[c] = constants,E[aX+b] =aE[X] + Y, thenE[X] E[Y]Now let s think aboutE[X+Y].E[X+Y] = x,y(x+y)Pr (X=x,Y=y)= x,yxPr (X=x,Y=y) + x,yyPr (X=x,Y=y)= xx yPr (X=x,Y=y) + yy xPr (X=x,Y=y)by total probability, xPr (X=x,Y=y) = Pr (X=x), likewise xPr (X=x,Y=y) =Pr (Y=y). So,E[X+Y] = xxPr (X=x) + yyPr (Y=y)=E[X] +E[Y]Notice thatE[X] works just like a mean; in fact we can think of it as beingthe population mean (as opposed to the sample mean).The variance is the expectation of (X E[X]) (X) = xp(x)(x E[X])21which we can show isE[X2] (E[X]) (X) =E[(X E[X])2]=E[X2 2XE[X] + (E[X])2]=E[X2] E[2XE[X]] +E[(E[X])2]NowE[X] is just another constant, soE[(E[X])2]= (E[X])2, andE[2XE[X]] =2E[X]E[X] = 2(E[X])2.
= E[X]+E[Y] Notice that E[X] works just like a mean; in fact we can think of it as being the population mean (as opposed to the sample mean). The variance is the expectation of (X −E[X])2. Var(X) = X x p(x)(x−E…
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