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CondExpectation-4

CondExpectation-4 - Viva La Correlacin Say X and Y are...

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1 Viva La Correlación! Say X and Y are arbitrary random variables Correlation of X and Y, denoted r (X, Y) : Note: -1  r (X, Y) 1 Correlation measures linearity between X and Y r (X, Y) = 1 Y = aX + b where a = s y / s x r (X, Y) = -1 Y = aX + b where a = - s y / s x r (X, Y) = 0 absence of linear relationship o But, X and Y can still be related in some other way! If r (X, Y) = 0, we say X and Y are “uncorrelated” o Note: Independence implies uncorrelated, but not vice versa! Y) Var(X)Var( ) , ( Cov ) , ( Y X Y X r Fun with Indicator Variables Let I A and I B be indicators for events A and B E[ I A ] = P(A), E[ I B ] = P(B), E[ I A I B ] = P(AB) Cov( I A , I B ) = E[ I A I B ] E[ I A ] E[ I B ] = P(AB) P(A)P(B) = P(A | B)P(B) P(A)P(B) = P(B)[P(A | B) P(A)] Cov( I A , I B ) determined by P(A | B) P(A) P(A | B) > P(A) r ( I A , I B ) > 0 P(A | B) = P(A) r ( I A , I B ) = 0 (and Cov( I A , I B ) = 0) P(A | B) < P(A) r ( I A , I B ) < 0 otherwise 0 occurs if 1 A I A otherwise 0 occurs if 1 B I B Can’t Get Enough of that Multinomial Multinomial distribution n independent trials of experiment performed Each trials results in one of m outcomes, with respective probabilities: p 1 , p 2 , …, p m where X i = number of trials with outcome i E.g., Rolling 6-sided die multiple times and counting how many of each value {1, 2, 3, 4, 5, 6} we get Would expect that X i are negatively correlated Let’s see... when i j , what is Cov( X i , X j )? m i i p 1 1 m c m c c m m m p p p c c c n c X c X c X P ... ,..., , ) ,..., , ( 2 1 2 1 2 1 2 2 1 1 Covariance and the Multinomial Computing Cov( X i , X j ) Indicator I i ( k ) = 1 if trial k has outcome i , 0 otherwise When a b , trial a and b independent: When a = b : Since trial a cannot have outcome i and
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