Lecture 9/Conditional Expectations1/ 3rd Stage / 2nd Course /2018-2019

We will discuss the properties of conditional expectation in more detail as they are quite useful in practice. We will also discuss conditional variance. An important concept here is that we interpret the conditional expectation as a random variable.
Remember that the conditional expectation of XX given that Y=yY=y is given by
E[X|Y=y]=?xi?RXxiPX|Y(xi|y).E[X|Y=y]=?xi?RXxiPX|Y(xi|y).
Note that E[X|Y=y]E[X|Y=y] depends on the value of yy. In other words, by changing yy, E[X|Y=y]E[X|Y=y]can also change. Thus, we can say E[X|Y=y]E[X|Y=y] is a function of yy, so let s write
g(y)=E[X|Y=y].g(y)=E[X|Y=y].
Thus, we can think of g(y)=E[X|Y=y]g(y)=E[X|Y=y] as a function of the value of random variable YY. We then write
g(Y)=E[X|Y].g(Y)=E[X|Y].
We use this notation to indicate that E[X|Y]E[X|Y] is a random variable whose value equals g(y)=E[X|Y=y]g(y)=E[X|Y=y] when Y=yY=y. Thus, if YY is a random variable with range RY=RY={y1,y2,?}{y1,y2,?},
probability P(Y=y2)......E[X|Y]={E[X|Y=y1]with probability P(Y=y1)E[X|Y=y2]with probability P(Y=y2)......
Let s look at an example.
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Example Let X=aY+bX=aY+b. Then E[X|Y=y]=E[aY+b|Y=y]=ay+bE[X|Y=y]=E[aY+b|Y=y]=ay+b. Here, we have g(y)=ay+bg(y)=ay+b, and therefore,
E[X|Y]=aY+b,E[X|Y]=aY+b,
which is a function of the random variable YY.
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Since E[X|Y]E[X|Y] is a random variable, we can find its PMF, CDF, variance, etc. Let s look at an example to better understand E[X|Y]E[X|Y].
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Example Consider two random variables XX and YY with joint PMF given in Table 5.2. Let Z=E[X|Y]Z=E[X|Y].
a. Find the Marginal PMFs of XX and YY.
b. Find the conditional PMF of XX given Y=0Y=0 and Y=1Y=1, i.e., find PX|Y(x|0)PX|Y(x|0) and PX|Y(x|1)PX|Y(x|1).
c. Find the PMFPMF of ZZ.
d. Find EZEZ, and check that EZ=EXEZ=EX.
e. Find Var(Z)(Z).