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).
4 Geometric random variables Suppose we keep trying independent Bernoulli variables until we have a success; each has probability of success p. Then the probability that the number of failures is k is (1−p)kp. (Be careful, some people use p as the probability of failure here, i.e. they reverse p and 1−p.)
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