STAT/MATH 309 DISCUSSION 9
TA: Jingjiang(Jack) Peng
Office: 1275 MSC, 1300 Universtiy Avenue
E-mail: [email protected]
Phone: 262-1577
Office Hour: 11:30-1:30 p.m. Tuesday or by appoitment
Website: www.stat.wisc.edu/ā¼peng
1 Conditional Expectation
ā¢If X is discrete, then the conditional expectation of X, given an event A, is equal to
E(X|A) = PxP (X=x|A)
ā¢If X and Y are discrete, then E(X|Y) is a random variable, with E(X|Y) equal to
E(X|Y=y) when Y=y
ā¢If X and Y are jointly continous, then E(X|Y=y) = RxfX|Y(x|y)dx, and again
E(X|Y) is a random variable, with E(X|Y) equal to E(X|Y=y) when Y=y
ā¢Conditional expectation is linear, i.e. E(aX1+bX2|Y) = aE(X1|Y) + bE(X2|Y)
ā¢Law of total expectation: E(E(X|Y)) = E(X)
ā¢E[g(Y)E(X|Y)] = E[g(Y)X], E[g(Y)|Y] = g(Y), E[E(X|Y)|Y] = E(X|Y)
ā¢Conditional Variance formula V ar(X|Y) = E(X2|Y)ā(E(X|Y))2
ā¢Calculate Variance based on conditioning. V ar(X) = V ar[E(X|Y)] + E[V ar(X|Y)]
2 Inequalities
ā¢For nonnegative X, Markovās Inequality says P(Xgeqa)ā¤EX
a
ā¢Chebychevās Inequality says P(|YāĀµY| ā„ a)ā¤V ar(Y)
a2
ā¢The Cauchy-Schwartz Inequality says |Cov(X , Y )| ā¤ (V ar(X)V ar(Y))1/2. So |Corr(X, Y )| ā¤
1
ā¢Jensenās Inequality f(E(X)) ā¤E(f(X)) whenever fis convex.
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