1. ELEC 303, Koushanfar, Fall’09 ELEC 303 – Random Signals Lecture 12 – More on conditional expectation and variance Dr. Farinaz Koushanfar ECE Dept., Rice University Oct 6, 2009

2. ELEC 303, Koushanfar, Fall’09 Lecture outline Reading: 4.3-4.4 Law of iterated expectations Law of total variance

3. ELEC 303, Koushanfar, Fall’09 Iterated expectations E[X|Y] is a function of Y Example: biased coin, prob(head)=Y~PDF[0,1] – For n coin tosses record X, the number of heads – For any Y  [0,1]  E[X|Y=y]=ny, so E[X|Y] is a RV Since this is a RV, it has an expectation: As long as X has a well-defined expectation: Law of Iterated expectations: E[E[X|Y]]=E[X] For any function g we have: E[Xg(Y)|Y]=g(Y)E[X|Y]

4. ELEC 303, Koushanfar, Fall’09 Example: law of iterated expectations We have a stick with length l Select a random point and break it once Keep the left piece and break it again Expected length of the remaining piece?

5. ELEC 303, Koushanfar, Fall’09 Example: forecast review X: sales of a company over the entire year Y: sales in the first sem of a coming year Assume the joint distribution of X,Y is known The E[X] serves as the forecast of the actual sale in the beginning of the year After the mid year, Y is known  E[X|Y] Forecast revision is: E[X|Y]-E[X] Find the expected value of the forecast revision

6. ELEC 303, Koushanfar, Fall’09 Conditional expectation as an estimator Y can be observations providing info about X The conditional expectation: The estimation error:  Thus, the estimation error does not have a systematic upward or downward bias

7. ELEC 303, Koushanfar, Fall’09 Conditional expectation as an estimator (cont’d) Is there correlation? Between Thus, An important property is that

8. ELEC 303, Koushanfar, Fall’09 Law of total variance This is a function of Y with: Use the mean and the law of iterated means Now, rewrite Law of total variance var(X)=E[var(X|Y)]+var(E[X|Y])

9. ELEC 303, Koushanfar, Fall’09 Law of total variance: example 1 N independent tosses of a biased coin Prob(head0=Y~U[0,1] X is the number of obtained heads Use law of total variance to find var(X)

10. ELEC 303, Koushanfar, Fall’09 Law of total variance: example 2 Consider breaking the stick twice ex. again Y: length of the stick after first break X: length after second break Use law of total variance to find var(X)

11. ELEC 303, Koushanfar, Fall’09 Law of total variance: example 3 Continuous RV X with the PDF: – f X (x)=1/2 (0≤x≤1) and f X (x)=1/4 (1≤x≤3) Define Y as: Y=1 for x<1, and Y=2 for x  1 Use law of total variance to find the variance