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Unformatted text preview: PUBH 7430 Lecture 2 Julian Wolfson Division of Biostatistics University of Minnesota School of Public Health September 8, 2011 Correlation and probability Independence and lack thereof Two events A and B are said to be independent if P ( A occurs and B occurs) = P ( A occurs) × P ( B occurs) Example Let Y 1 and Y 2 be the results (0=heads,1=tails) of flipping a fair coin twice. P (two heads) = P ( Y 1 = 0 , Y 2 = 0) = P ( Y 1 = 0) P ( Y 2 = 0) = 1 / 4 Independence and lack thereof We commonly refer to events A and B as “correlated” if they are not independent : P ( A occurs and B occurs) 6 = P ( A occurs) × P ( B occurs) Example Let Y 1 and Y 2 be the results (0=heads,1=tails) of the following coinflipping experiment: • Flip the coin once, and record Y 1 as the first flip • If the first flip is a tails, flip the coin again and record Y 2 as the second flip • If the first flip is heads, flip the coin two more times. Record Y 2 as heads if one or two heads come up, and as tails if no heads come up. P (two heads) = P ( Y 1 = 0 , Y 2 = 0) = 1 / 2 × 3 / 4 = 3 / 8 6 = 1 / 4 Consequences of correlation • Many statistics (eg. mean) are based on sums • An important task is to estimate the variance of a given statistic • How does correlation impact variance estimates? Let Y 1 and Y 2 be two random variables (possibly dependent). What is the variance of Y 1 + Y 2 ? Var ( Y 1 + Y 2 ) = Var ( Y 1 )+ Var ( Y 2 )+ 2[ E ( Y 1 Y 2 ) E ( Y 1 ) E ( Y 2 )] Covariance Var ( Y 1 + Y 2 ) = Var ( Y 1 )+ Var ( Y 1 )+ 2[ E ( Y 1 Y 2 ) E ( Y 1 ) E ( Y 2 )] We give the part in red a special name, the covariance : Cov ( X , Y ) = E ( XY ) E ( X ) E ( Y ) Covariance – properties Cov ( X , Y ) = E ( XY ) E ( X ) E ( Y ) ≡ E ( X E ( X )) E ( Y E ( Y )) • Cov ( X , X ) = Var ( X ) • Covariance is symmetric : Cov ( X , Y ) = Cov ( Y , X ) • Covariance is unscaled : Cov (2 X , 3 Y ) 6 = Cov ( X , Y ) Covariance – properties Recall: If Y 1 and Y 2 are independent , we have P ( Y 1 = y 1 , Y 2 = y 2 ) = P ( Y 1 = y 1 ) × P ( Y 2 = y 2 ) One can show that independence also implies that E ( Y 1 Y 2 ) = E ( Y 1 ) × E ( Y 2 ), and hence... Cov ( Y 1 , Y 2 ) = E ( Y 1 Y 2 ) E ( Y 1 ) E ( Y 2 ) = E ( Y 1 ) E ( Y 2 ) E ( Y 1 ) E ( Y 2 ) = 0 (Note: Cov ( Y 1 , Y 2 ) = 0 does not necessarily imply that Y 1 and Y 2 are independent) Consequences of correlation More generally, Var ( Y 1 + ··· + Y n ) = n X i =1 Var ( Y i...
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 Fall '04
 Prof.Eberly

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