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ELEC 303 – Random Signals

ELEC 303 – Random Signals. Lecture 12 – More on conditional expectation and variance Dr. Farinaz Koushanfar ECE Dept., Rice University Oct 6, 2009. Lecture outline. Reading: 4.3-4.4 Law of iterated expectations Law of total variance. Iterated expectations. E[X|Y] is a function of Y

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ELEC 303 – Random Signals

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

  2. Lecture outline Reading: 4.3-4.4 Law of iterated expectations Law of total variance

  3. 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. 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. 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. 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. Conditional expectation as an estimator (cont’d) Is there correlation? Between Thus, An important property is that

  8. 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. 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. 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. Law of total variance: example 3 • Continuous RV X with the PDF: • fX(x)=1/2 (0≤x≤1) and fX(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

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