EE 342 4.3.17 - On Covariance; Correlation coefficient; Conditional Expectation; Law of Iterated PDF

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On Covariance; Correlation coefficient; Conditional Expectation; Law of Iterated Expectations...

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Monday, April 3, 2017

EE 342 Announcements

- PS10 (HW) due Friday, 4/7 - Midterm 2 next week Wednesday, 4/12 • On HW (6-10) and things covered in class - Prof recommends solving HW again; writing down the formula before solving if not completely sure how to answer the question

- Final Exam will be cumulative

Covariance cov(x,y) = E[(x-E[x])(y -E[y])] = E[XY] - E[X] E[Y] cov (x,y) = 0 => uncorrelated cov (x,y) > 0 = positive correlation cov (x,y) < 0 = negative correlation

X, Y independent => cov(X,Y) = 0, also uncorrelated

- Independent implies uncorrelated, but uncorrelated doesn’t necessarily imply independent (AKA: the converse is not true)

Correlation coefficient cov(x,y) > cov(x,z)

- cannot really compare, have to nomalize to compare

∫(x,y) = cov(x,y) / √var(x)var(y) 1

Monday, April 3, 2017 -1≤ ∫ E[z] = E[E(X|Y)] = E[g(Y)] = Σy g(y) PY (y) ∫ g(y) fY(y) dy = Σy E[X|Y] PY(y) ∫ E[X|Y] fY(y) dy = E[X] = Σy E[x|y]PY(y) = ΣyΣx x PX|Y(x|y)PY(y) = ΣyΣx x PX,Y(x,y) 2

Monday, April 3, 2017 = Σx x Σy PXY (x,y) = Σx x PX(x) = E[X]

Ex: Biased coin, where possibility of Head is random variable Y

- Y = probability of H, uniform [0,1] - After n tosses, get X = # of H - What is the expected number of H? (AKA: E[X] = ?) • If Y=y is known, would be E[X|Y=y] = ny • When Y=y is not known, E[X|Y] = nY

E[X] = E[E[X|Y]] = E[nY] = nE[Y] = n/2

- same result even with a non-biased coin

Ex: (HW9,Problem#2) E[X] = ? If Y=y, X is uniform [0,y] E[X|Y=y] = y/2 E[X|Y] = Y/2 E[X] = E[E[X|Y]]= E[Y/2] = 1/2 E[Y] = l/4

3...